NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.6%

Time: 18.3s

Alternatives: 19

Speedup: 1.9×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2e-134)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- -1.0 eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-134) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2e-134)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2e-134], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.00000000000000008e-134

   1. Initial program 64.4%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in eps around 0 31.7%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in eps around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. +-commutative 67.4%

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

      3. *-commutative 67.4%

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

      4. fma-undefine 67.4%

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

      5. rec-exp 67.4%

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

      6. associate-*r/ 67.5%

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

      7. *-rgt-identity 67.5%

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

   10. Simplified 67.5%

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

   if 2.00000000000000008e-134 < eps

   1. Initial program 85.2%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. exp-sum 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-lft-identity 100.0%

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

      5. exp-sum 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. rec-exp 100.0%

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

      9. fma-undefine 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-57)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-57) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-57)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5e-57], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.0000000000000002e-57

   1. Initial program 61.8%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in eps around 0 31.4%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in eps around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. +-commutative 69.7%

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

      3. *-commutative 69.7%

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

      4. fma-undefine 69.7%

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

      5. rec-exp 69.7%

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

      6. associate-*r/ 69.8%

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

      7. *-rgt-identity 69.8%

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

   10. Simplified 69.8%

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

   if 5.0000000000000002e-57 < eps

   1. Initial program 93.2%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. exp-sum 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-lft-identity 100.0%

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

      5. exp-sum 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. rec-exp 100.0%

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

      9. fma-undefine 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right) - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -8.5e-247)
   (/ (+ 2.0 (* x (- (* 0.5 (* x (pow (+ eps_m -1.0) 2.0))) 2.0))) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (- x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -8.5e-247) {
		tmp = (2.0 + (x * ((0.5 * (x * pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-8.5d-247)) then
        tmp = (2.0d0 + (x * ((0.5d0 * (x * ((eps_m + (-1.0d0)) ** 2.0d0))) - 2.0d0))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -8.5e-247) {
		tmp = (2.0 + (x * ((0.5 * (x * Math.pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -8.5e-247:
		tmp = (2.0 + (x * ((0.5 * (x * math.pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -8.5e-247)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0))) - 2.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -8.5e-247)
		tmp = (2.0 + (x * ((0.5 * (x * ((eps_m + -1.0) ^ 2.0))) - 2.0))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -8.5e-247], N[(N[(2.0 + N[(x * N[(N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000003e-247

   1. Initial program 71.3%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in eps around inf 95.4%

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

   6. Taylor expanded in x around 0 69.0%

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

      1. mul-1-neg 69.0%

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

   8. Simplified 69.0%

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

   9. Taylor expanded in x around 0 86.7%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right) - 2\right)}}{2} \]

   if -8.5000000000000003e-247 < x

   1. Initial program 72.9%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around 0 83.9%

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

      1. rec-exp 83.9%

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

   8. Simplified 83.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right) - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right) - 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 8.2e-14)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (+ 2.0 (* x (- (* 0.5 (* x (pow (+ eps_m -1.0) 2.0))) 2.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8.2e-14) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((0.5 * (x * pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 8.2e-14)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0))) - 2.0))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 8.2e-14], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.2000000000000004e-14

   1. Initial program 60.0%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in eps around 0 30.8%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in eps around 0 70.9%

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

      1. *-commutative 70.9%

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

      2. +-commutative 70.9%

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

      3. *-commutative 70.9%

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

      4. fma-undefine 70.9%

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

      5. rec-exp 70.9%

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

      6. associate-*r/ 71.0%

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

      7. *-rgt-identity 71.0%

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

   10. Simplified 71.0%

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

   if 8.2000000000000004e-14 < eps

   1. Initial program 100.0%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in x around 0 83.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right) - 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 3 \cdot 10^{+14} \lor \neg \left(eps\_m \leq 2.3 \cdot 10^{+222}\right) \land eps\_m \leq 2.8 \cdot 10^{+235}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= eps_m 3e+14) (and (not (<= eps_m 2.3e+222)) (<= eps_m 2.8e+235)))
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((eps_m <= 3e+14) || (!(eps_m <= 2.3e+222) && (eps_m <= 2.8e+235))) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((eps_m <= 3d+14) .or. (.not. (eps_m <= 2.3d+222)) .and. (eps_m <= 2.8d+235)) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((eps_m <= 3e+14) || (!(eps_m <= 2.3e+222) && (eps_m <= 2.8e+235))) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (eps_m <= 3e+14) or (not (eps_m <= 2.3e+222) and (eps_m <= 2.8e+235)):
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((eps_m <= 3e+14) || (!(eps_m <= 2.3e+222) && (eps_m <= 2.8e+235)))
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((eps_m <= 3e+14) || (~((eps_m <= 2.3e+222)) && (eps_m <= 2.8e+235)))
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[eps$95$m, 3e+14], And[N[Not[LessEqual[eps$95$m, 2.3e+222]], $MachinePrecision], LessEqual[eps$95$m, 2.8e+235]]], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3e14 or 2.30000000000000011e222 < eps < 2.80000000000000026e235

   1. Initial program 62.3%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in eps around inf 97.0%

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

   6. Taylor expanded in eps around 0 82.4%

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

      1. rec-exp 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in eps around 0 75.0%

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

       1. neg-mul-1 75.0%

          \[\leadsto \frac{e^{\color{blue}{-1 \cdot x}} + e^{-1 \cdot x}}{2} \]

       2. count-2 75.0%

          \[\leadsto \frac{\color{blue}{2 \cdot e^{-1 \cdot x}}}{2} \]

       3. neg-mul-1 75.0%

          \[\leadsto \frac{2 \cdot e^{\color{blue}{-x}}}{2} \]

   11. Simplified 75.0%

       \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]

   if 3e14 < eps < 2.30000000000000011e222 or 2.80000000000000026e235 < eps

   1. Initial program 100.0%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 71.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3 \cdot 10^{+14} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{+222}\right) \land \varepsilon \leq 2.8 \cdot 10^{+235}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 9500000000000:\\ \;\;\;\;\frac{t\_0 \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 2.3 \cdot 10^{+222} \lor \neg \left(eps\_m \leq 2.8 \cdot 10^{+235}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 9500000000000.0)
     (/ (* t_0 (+ 2.0 (* x 2.0))) 2.0)
     (if (or (<= eps_m 2.3e+222) (not (<= eps_m 2.8e+235)))
       (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
       (/ (* 2.0 t_0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 9500000000000.0) {
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0;
	} else if ((eps_m <= 2.3e+222) || !(eps_m <= 2.8e+235)) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = (2.0 * t_0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 9500000000000.0d0) then
        tmp = (t_0 * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else if ((eps_m <= 2.3d+222) .or. (.not. (eps_m <= 2.8d+235))) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = (2.0d0 * t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 9500000000000.0) {
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0;
	} else if ((eps_m <= 2.3e+222) || !(eps_m <= 2.8e+235)) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = (2.0 * t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 9500000000000.0:
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0
	elif (eps_m <= 2.3e+222) or not (eps_m <= 2.8e+235):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = (2.0 * t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 9500000000000.0)
		tmp = Float64(Float64(t_0 * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif ((eps_m <= 2.3e+222) || !(eps_m <= 2.8e+235))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 9500000000000.0)
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0;
	elseif ((eps_m <= 2.3e+222) || ~((eps_m <= 2.8e+235)))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = (2.0 * t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 9500000000000.0], N[(N[(t$95$0 * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps$95$m, 2.3e+222], N[Not[LessEqual[eps$95$m, 2.8e+235]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 9.5e12

   1. Initial program 61.1%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in eps around 0 31.6%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in eps around 0 71.7%

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

   if 9.5e12 < eps < 2.30000000000000011e222 or 2.80000000000000026e235 < eps

   1. Initial program 100.0%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 71.1%

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

   if 2.30000000000000011e222 < eps < 2.80000000000000026e235

   1. Initial program 100.0%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 49.2%

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

      1. rec-exp 49.2%

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

   8. Simplified 49.2%

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

   9. Taylor expanded in eps around 0 51.6%

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

       1. neg-mul-1 51.6%

          \[\leadsto \frac{e^{\color{blue}{-1 \cdot x}} + e^{-1 \cdot x}}{2} \]

       2. count-2 51.6%

          \[\leadsto \frac{\color{blue}{2 \cdot e^{-1 \cdot x}}}{2} \]

       3. neg-mul-1 51.6%

          \[\leadsto \frac{2 \cdot e^{\color{blue}{-x}}}{2} \]

   11. Simplified 51.6%

       \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 9500000000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{+222} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{+235}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right) - 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 8.2e-14)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (/ (+ 2.0 (* x (- (* 0.5 (* x (pow (+ eps_m -1.0) 2.0))) 2.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8.2e-14) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((0.5 * (x * pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 8.2d-14) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((0.5d0 * (x * ((eps_m + (-1.0d0)) ** 2.0d0))) - 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8.2e-14) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((0.5 * (x * Math.pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 8.2e-14:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (2.0 + (x * ((0.5 * (x * math.pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 8.2e-14)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0))) - 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 8.2e-14)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (2.0 + (x * ((0.5 * (x * ((eps_m + -1.0) ^ 2.0))) - 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 8.2e-14], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.2000000000000004e-14

   1. Initial program 60.0%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in eps around 0 30.8%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in eps around 0 70.9%

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

   if 8.2000000000000004e-14 < eps

   1. Initial program 100.0%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in x around 0 83.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right) - 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right) - 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 8.2e-14)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ 2.0 (* x (- (* 0.5 (* x (pow (+ eps_m -1.0) 2.0))) 2.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8.2e-14) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (2.0 + (x * ((0.5 * (x * pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 8.2d-14) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((0.5d0 * (x * ((eps_m + (-1.0d0)) ** 2.0d0))) - 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8.2e-14) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (2.0 + (x * ((0.5 * (x * Math.pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 8.2e-14:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (2.0 + (x * ((0.5 * (x * math.pow((eps_m + -1.0), 2.0))) - 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 8.2e-14)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0))) - 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 8.2e-14)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (2.0 + (x * ((0.5 * (x * ((eps_m + -1.0) ^ 2.0))) - 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 8.2e-14], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.2000000000000004e-14

   1. Initial program 60.0%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in eps around 0 30.8%

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

   7. Simplified 70.9%

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

   if 8.2000000000000004e-14 < eps

   1. Initial program 100.0%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in x around 0 83.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right) - 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 22000000000:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;eps\_m \leq 6.2 \cdot 10^{+214}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 22000000000.0)
   (/ (* 2.0 (exp (- x))) 2.0)
   (if (<= eps_m 6.2e+214)
     (/ (* (exp x) (+ x (+ x 2.0))) 2.0)
     (/
      (* x (+ 2.0 (* x (+ 2.0 (* x (+ 1.0 (* x 0.3333333333333333)))))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 22000000000.0) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if (eps_m <= 6.2e+214) {
		tmp = (exp(x) * (x + (x + 2.0))) / 2.0;
	} else {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 22000000000.0d0) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if (eps_m <= 6.2d+214) then
        tmp = (exp(x) * (x + (x + 2.0d0))) / 2.0d0
    else
        tmp = (x * (2.0d0 + (x * (2.0d0 + (x * (1.0d0 + (x * 0.3333333333333333d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 22000000000.0) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if (eps_m <= 6.2e+214) {
		tmp = (Math.exp(x) * (x + (x + 2.0))) / 2.0;
	} else {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 22000000000.0:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif eps_m <= 6.2e+214:
		tmp = (math.exp(x) * (x + (x + 2.0))) / 2.0
	else:
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 22000000000.0)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif (eps_m <= 6.2e+214)
		tmp = Float64(Float64(exp(x) * Float64(x + Float64(x + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.3333333333333333))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 22000000000.0)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif (eps_m <= 6.2e+214)
		tmp = (exp(x) * (x + (x + 2.0))) / 2.0;
	else
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 22000000000.0], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 6.2e+214], N[(N[(N[Exp[x], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(2.0 + N[(x * N[(2.0 + N[(x * N[(1.0 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 2.2e10

   1. Initial program 60.9%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in eps around inf 96.9%

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

   6. Taylor expanded in eps around 0 83.4%

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

      1. rec-exp 83.4%

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

   8. Simplified 83.4%

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

   9. Taylor expanded in eps around 0 75.6%

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

       1. neg-mul-1 75.6%

          \[\leadsto \frac{e^{\color{blue}{-1 \cdot x}} + e^{-1 \cdot x}}{2} \]

       2. count-2 75.6%

          \[\leadsto \frac{\color{blue}{2 \cdot e^{-1 \cdot x}}}{2} \]

       3. neg-mul-1 75.6%

          \[\leadsto \frac{2 \cdot e^{\color{blue}{-x}}}{2} \]

   11. Simplified 75.6%

       \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]

   if 2.2e10 < eps < 6.19999999999999957e214

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 34.7%

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

      1. associate--r+ 34.7%

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

      2. associate-*r* 34.7%

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

      3. mul-1-neg 34.7%

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

      4. cancel-sign-sub 34.7%

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

      5. distribute-rgt1-in 34.7%

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

      6. distribute-rgt-out-- 34.7%

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

      7. mul-1-neg 34.7%

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

      8. mul-1-neg 34.7%

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

   7. Simplified 34.7%

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

      1. *-commutative 34.7%

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

      2. distribute-lft-out 34.7%

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

      3. add-sqr-sqrt 17.0%

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

      4. sqrt-unprod 68.2%

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

      5. sqr-neg 68.2%

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

      6. sqrt-unprod 51.1%

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

      7. add-sqr-sqrt 68.5%

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

      8. associate--l+ 68.5%

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

      9. metadata-eval 68.5%

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

   9. Applied egg-rr 68.5%

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

   if 6.19999999999999957e214 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 6.3%

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

      1. associate--r+ 6.3%

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

      2. associate-*r* 6.3%

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

      3. mul-1-neg 6.3%

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

      4. cancel-sign-sub 6.3%

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

      5. distribute-rgt1-in 6.3%

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

      6. distribute-rgt-out-- 6.3%

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

      7. mul-1-neg 6.3%

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

      8. mul-1-neg 6.3%

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

   7. Simplified 6.3%

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

      1. *-commutative 6.3%

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

      2. distribute-lft-out 6.3%

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

      3. add-sqr-sqrt 5.3%

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

      4. sqrt-unprod 28.7%

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

      5. sqr-neg 28.7%

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

      6. sqrt-unprod 23.4%

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

      7. add-sqr-sqrt 29.1%

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

      8. associate--l+ 29.1%

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

      9. metadata-eval 29.1%

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

   9. Applied egg-rr 29.1%

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

   10. Taylor expanded in x around inf 24.0%

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

       1. *-commutative 24.0%

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

       2. *-commutative 24.0%

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

       3. associate-*r* 24.0%

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

   12. Simplified 24.0%

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

   13. Taylor expanded in x around 0 42.1%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + 0.3333333333333333 \cdot x\right)\right)\right)}}{2} \]
   14. Step-by-step derivation

       1. *-commutative 42.1%

          \[\leadsto \frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + \color{blue}{x \cdot 0.3333333333333333}\right)\right)\right)}{2} \]

   15. Simplified 42.1%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 22000000000:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{+214}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.2e+77)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (* x (+ 2.0 (* x (+ 2.0 (* x (+ 1.0 (* x 0.3333333333333333))))))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2e+77) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.2d+77) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (x * (2.0d0 + (x * (2.0d0 + (x * (1.0d0 + (x * 0.3333333333333333d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2e+77) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.2e+77:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.2e+77)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.3333333333333333))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.2e+77)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.2e+77], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(2.0 + N[(x * N[(2.0 + N[(x * N[(1.0 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e77

   1. Initial program 65.1%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in eps around inf 97.2%

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

   6. Taylor expanded in eps around 0 88.0%

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

      1. rec-exp 88.0%

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

   8. Simplified 88.0%

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

   9. Taylor expanded in eps around 0 76.0%

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

       1. neg-mul-1 76.0%

          \[\leadsto \frac{e^{\color{blue}{-1 \cdot x}} + e^{-1 \cdot x}}{2} \]

       2. count-2 76.0%

          \[\leadsto \frac{\color{blue}{2 \cdot e^{-1 \cdot x}}}{2} \]

       3. neg-mul-1 76.0%

          \[\leadsto \frac{2 \cdot e^{\color{blue}{-x}}}{2} \]

   11. Simplified 76.0%

       \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]

   if 2.2e77 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 40.6%

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

      1. associate--r+ 40.6%

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

      2. associate-*r* 40.6%

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

      3. mul-1-neg 40.6%

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

      4. cancel-sign-sub 40.6%

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

      5. distribute-rgt1-in 40.6%

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

      6. distribute-rgt-out-- 40.6%

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

      7. mul-1-neg 40.6%

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

      8. mul-1-neg 40.6%

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

   7. Simplified 40.6%

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

      1. *-commutative 40.6%

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

      2. distribute-lft-out 40.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 61.0%

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

      5. sqr-neg 61.0%

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

      6. sqrt-unprod 61.0%

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

      7. add-sqr-sqrt 61.0%

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

      8. associate--l+ 61.0%

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

      9. metadata-eval 61.0%

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

   9. Applied egg-rr 61.0%

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

   10. Taylor expanded in x around inf 61.0%

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

       1. *-commutative 61.0%

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

       2. *-commutative 61.0%

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

       3. associate-*r* 61.0%

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

   12. Simplified 61.0%

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

   13. Taylor expanded in x around 0 61.0%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + 0.3333333333333333 \cdot x\right)\right)\right)}}{2} \]
   14. Step-by-step derivation

       1. *-commutative 61.0%

          \[\leadsto \frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + \color{blue}{x \cdot 0.3333333333333333}\right)\right)\right)}{2} \]

   15. Simplified 61.0%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{if}\;x \leq -550:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 2.0 (* x 2.0))) 2.0)))
   (if (<= x -550.0)
     t_0
     (if (<= x 480.0)
       1.0
       (if (<= x 9.5e+77)
         0.0
         (if (<= x 1.5e+131)
           (/ (* x (+ 2.0 (* x (+ x 2.0)))) 2.0)
           (if (<= x 1.6e+207) 0.0 t_0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -550.0) {
		tmp = t_0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 9.5e+77) {
		tmp = 0.0;
	} else if (x <= 1.5e+131) {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	} else if (x <= 1.6e+207) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (2.0d0 + (x * 2.0d0))) / 2.0d0
    if (x <= (-550.0d0)) then
        tmp = t_0
    else if (x <= 480.0d0) then
        tmp = 1.0d0
    else if (x <= 9.5d+77) then
        tmp = 0.0d0
    else if (x <= 1.5d+131) then
        tmp = (x * (2.0d0 + (x * (x + 2.0d0)))) / 2.0d0
    else if (x <= 1.6d+207) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -550.0) {
		tmp = t_0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 9.5e+77) {
		tmp = 0.0;
	} else if (x <= 1.5e+131) {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	} else if (x <= 1.6e+207) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x * (2.0 + (x * 2.0))) / 2.0
	tmp = 0
	if x <= -550.0:
		tmp = t_0
	elif x <= 480.0:
		tmp = 1.0
	elif x <= 9.5e+77:
		tmp = 0.0
	elif x <= 1.5e+131:
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0
	elif x <= 1.6e+207:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x * Float64(2.0 + Float64(x * 2.0))) / 2.0)
	tmp = 0.0
	if (x <= -550.0)
		tmp = t_0;
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 9.5e+77)
		tmp = 0.0;
	elseif (x <= 1.5e+131)
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + 2.0)))) / 2.0);
	elseif (x <= 1.6e+207)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	tmp = 0.0;
	if (x <= -550.0)
		tmp = t_0;
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 9.5e+77)
		tmp = 0.0;
	elseif (x <= 1.5e+131)
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	elseif (x <= 1.6e+207)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -550.0], t$95$0, If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 9.5e+77], 0.0, If[LessEqual[x, 1.5e+131], N[(N[(x * N[(2.0 + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+207], 0.0, t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -550 or 1.6000000000000001e207 < x

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in eps around 0 13.2%

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

      1. associate--r+ 13.2%

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

      2. associate-*r* 13.2%

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

      3. mul-1-neg 13.2%

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

      4. cancel-sign-sub 13.2%

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

      5. distribute-rgt1-in 13.2%

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

      6. distribute-rgt-out-- 15.0%

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

      7. mul-1-neg 15.0%

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

      8. mul-1-neg 15.0%

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

   7. Simplified 15.0%

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

      1. *-commutative 15.0%

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

      2. distribute-lft-out 15.0%

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

      3. add-sqr-sqrt 1.8%

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

      4. sqrt-unprod 29.3%

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

      5. sqr-neg 29.3%

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

      6. sqrt-unprod 27.5%

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

      7. add-sqr-sqrt 28.4%

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

      8. associate--l+ 28.4%

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

      9. metadata-eval 28.4%

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

   9. Applied egg-rr 28.4%

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

   10. Taylor expanded in x around inf 28.4%

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

       1. *-commutative 28.4%

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

       2. *-commutative 28.4%

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

       3. associate-*r* 28.4%

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

   12. Simplified 28.4%

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

   13. Taylor expanded in x around 0 64.8%

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

       1. *-commutative 64.8%

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

   15. Simplified 64.8%

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

   if -550 < x < 480

   1. Initial program 52.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in x around 0 72.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 480 < x < 9.4999999999999998e77 or 1.5000000000000001e131 < x < 1.6000000000000001e207

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 69.5%

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

      1. mul-1-neg 69.5%

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

      2. mul-1-neg 69.5%

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

      3. rec-exp 69.5%

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

      4. sub-neg 69.5%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 69.5%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 69.5%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 69.5%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 69.5%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 69.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 9.4999999999999998e77 < x < 1.5000000000000001e131

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 16.7%

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

      1. associate--r+ 16.7%

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

      2. associate-*r* 16.7%

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

      3. mul-1-neg 16.7%

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

      4. cancel-sign-sub 16.7%

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

      5. distribute-rgt1-in 16.7%

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

      6. distribute-rgt-out-- 16.7%

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

      7. mul-1-neg 16.7%

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

      8. mul-1-neg 16.7%

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

   7. Simplified 16.7%

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

      1. *-commutative 16.7%

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

      2. distribute-lft-out 16.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 84.9%

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

      5. sqr-neg 84.9%

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

      6. sqrt-unprod 84.9%

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

      7. add-sqr-sqrt 84.9%

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

      8. associate--l+ 84.9%

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

      9. metadata-eval 84.9%

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

   9. Applied egg-rr 84.9%

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

   10. Taylor expanded in x around inf 84.9%

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

       1. *-commutative 84.9%

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

       2. *-commutative 84.9%

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

       3. associate-*r* 84.9%

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

   12. Simplified 84.9%

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

   13. Taylor expanded in x around 0 56.2%

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

       1. +-commutative 56.2%

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

   15. Simplified 56.2%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{if}\;x \leq 520:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 2.0 (* x (+ 4.0 (* x 3.0)))) 2.0)))
   (if (<= x 520.0)
     t_0
     (if (<= x 9.5e+77)
       0.0
       (if (<= x 3.5e+131)
         (/ (* x (+ 2.0 (* x (+ x 2.0)))) 2.0)
         (if (<= x 5e+207) 0.0 t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	double tmp;
	if (x <= 520.0) {
		tmp = t_0;
	} else if (x <= 9.5e+77) {
		tmp = 0.0;
	} else if (x <= 3.5e+131) {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	} else if (x <= 5e+207) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + (x * (4.0d0 + (x * 3.0d0)))) / 2.0d0
    if (x <= 520.0d0) then
        tmp = t_0
    else if (x <= 9.5d+77) then
        tmp = 0.0d0
    else if (x <= 3.5d+131) then
        tmp = (x * (2.0d0 + (x * (x + 2.0d0)))) / 2.0d0
    else if (x <= 5d+207) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	double tmp;
	if (x <= 520.0) {
		tmp = t_0;
	} else if (x <= 9.5e+77) {
		tmp = 0.0;
	} else if (x <= 3.5e+131) {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	} else if (x <= 5e+207) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0
	tmp = 0
	if x <= 520.0:
		tmp = t_0
	elif x <= 9.5e+77:
		tmp = 0.0
	elif x <= 3.5e+131:
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0
	elif x <= 5e+207:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 + Float64(x * Float64(4.0 + Float64(x * 3.0)))) / 2.0)
	tmp = 0.0
	if (x <= 520.0)
		tmp = t_0;
	elseif (x <= 9.5e+77)
		tmp = 0.0;
	elseif (x <= 3.5e+131)
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + 2.0)))) / 2.0);
	elseif (x <= 5e+207)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	tmp = 0.0;
	if (x <= 520.0)
		tmp = t_0;
	elseif (x <= 9.5e+77)
		tmp = 0.0;
	elseif (x <= 3.5e+131)
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	elseif (x <= 5e+207)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 + N[(x * N[(4.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 520.0], t$95$0, If[LessEqual[x, 9.5e+77], 0.0, If[LessEqual[x, 3.5e+131], N[(N[(x * N[(2.0 + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+207], 0.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 520 or 4.9999999999999999e207 < x

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in eps around 0 58.2%

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

      1. associate--r+ 58.2%

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

      2. associate-*r* 58.2%

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

      3. mul-1-neg 58.2%

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

      4. cancel-sign-sub 58.2%

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

      5. distribute-rgt1-in 58.2%

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

      6. distribute-rgt-out-- 58.7%

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

      7. mul-1-neg 58.7%

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

      8. mul-1-neg 58.7%

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

   7. Simplified 58.7%

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

      1. *-commutative 58.7%

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

      2. distribute-lft-out 58.7%

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

      3. add-sqr-sqrt 28.2%

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

      4. sqrt-unprod 61.8%

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

      5. sqr-neg 61.8%

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

      6. sqrt-unprod 33.6%

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

      7. add-sqr-sqrt 60.0%

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

      8. associate--l+ 60.0%

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

      9. metadata-eval 60.0%

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

   9. Applied egg-rr 60.0%

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

   10. Taylor expanded in x around 0 70.0%

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

       1. *-commutative 70.0%

          \[\leadsto \frac{2 + x \cdot \left(4 + \color{blue}{x \cdot 3}\right)}{2} \]

   12. Simplified 70.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(4 + x \cdot 3\right)}}{2} \]

   if 520 < x < 9.4999999999999998e77 or 3.4999999999999999e131 < x < 4.9999999999999999e207

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 69.5%

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

      1. mul-1-neg 69.5%

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

      2. mul-1-neg 69.5%

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

      3. rec-exp 69.5%

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

      4. sub-neg 69.5%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 69.5%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 69.5%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 69.5%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 69.5%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 69.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 9.4999999999999998e77 < x < 3.4999999999999999e131

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 16.7%

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

      1. associate--r+ 16.7%

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

      2. associate-*r* 16.7%

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

      3. mul-1-neg 16.7%

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

      4. cancel-sign-sub 16.7%

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

      5. distribute-rgt1-in 16.7%

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

      6. distribute-rgt-out-- 16.7%

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

      7. mul-1-neg 16.7%

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

      8. mul-1-neg 16.7%

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

   7. Simplified 16.7%

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

      1. *-commutative 16.7%

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

      2. distribute-lft-out 16.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 84.9%

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

      5. sqr-neg 84.9%

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

      6. sqrt-unprod 84.9%

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

      7. add-sqr-sqrt 84.9%

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

      8. associate--l+ 84.9%

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

      9. metadata-eval 84.9%

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

   9. Applied egg-rr 84.9%

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

   10. Taylor expanded in x around inf 84.9%

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

       1. *-commutative 84.9%

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

       2. *-commutative 84.9%

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

       3. associate-*r* 84.9%

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

   12. Simplified 84.9%

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

   13. Taylor expanded in x around 0 56.2%

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

       1. +-commutative 56.2%

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

   15. Simplified 56.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{if}\;x \leq 600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+131}:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot \left(3 + x \cdot 1.3333333333333333\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+205}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 2.0 (* x (+ 4.0 (* x 3.0)))) 2.0)))
   (if (<= x 600.0)
     t_0
     (if (<= x 9.5e+77)
       0.0
       (if (<= x 1e+131)
         (/ (+ 2.0 (* x (+ 4.0 (* x (+ 3.0 (* x 1.3333333333333333)))))) 2.0)
         (if (<= x 5e+205) 0.0 t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	double tmp;
	if (x <= 600.0) {
		tmp = t_0;
	} else if (x <= 9.5e+77) {
		tmp = 0.0;
	} else if (x <= 1e+131) {
		tmp = (2.0 + (x * (4.0 + (x * (3.0 + (x * 1.3333333333333333)))))) / 2.0;
	} else if (x <= 5e+205) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + (x * (4.0d0 + (x * 3.0d0)))) / 2.0d0
    if (x <= 600.0d0) then
        tmp = t_0
    else if (x <= 9.5d+77) then
        tmp = 0.0d0
    else if (x <= 1d+131) then
        tmp = (2.0d0 + (x * (4.0d0 + (x * (3.0d0 + (x * 1.3333333333333333d0)))))) / 2.0d0
    else if (x <= 5d+205) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	double tmp;
	if (x <= 600.0) {
		tmp = t_0;
	} else if (x <= 9.5e+77) {
		tmp = 0.0;
	} else if (x <= 1e+131) {
		tmp = (2.0 + (x * (4.0 + (x * (3.0 + (x * 1.3333333333333333)))))) / 2.0;
	} else if (x <= 5e+205) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0
	tmp = 0
	if x <= 600.0:
		tmp = t_0
	elif x <= 9.5e+77:
		tmp = 0.0
	elif x <= 1e+131:
		tmp = (2.0 + (x * (4.0 + (x * (3.0 + (x * 1.3333333333333333)))))) / 2.0
	elif x <= 5e+205:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 + Float64(x * Float64(4.0 + Float64(x * 3.0)))) / 2.0)
	tmp = 0.0
	if (x <= 600.0)
		tmp = t_0;
	elseif (x <= 9.5e+77)
		tmp = 0.0;
	elseif (x <= 1e+131)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(4.0 + Float64(x * Float64(3.0 + Float64(x * 1.3333333333333333)))))) / 2.0);
	elseif (x <= 5e+205)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	tmp = 0.0;
	if (x <= 600.0)
		tmp = t_0;
	elseif (x <= 9.5e+77)
		tmp = 0.0;
	elseif (x <= 1e+131)
		tmp = (2.0 + (x * (4.0 + (x * (3.0 + (x * 1.3333333333333333)))))) / 2.0;
	elseif (x <= 5e+205)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 + N[(x * N[(4.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 600.0], t$95$0, If[LessEqual[x, 9.5e+77], 0.0, If[LessEqual[x, 1e+131], N[(N[(2.0 + N[(x * N[(4.0 + N[(x * N[(3.0 + N[(x * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+205], 0.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 600 or 5.0000000000000002e205 < x

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in eps around 0 58.2%

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

      1. associate--r+ 58.2%

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

      2. associate-*r* 58.2%

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

      3. mul-1-neg 58.2%

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

      4. cancel-sign-sub 58.2%

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

      5. distribute-rgt1-in 58.2%

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

      6. distribute-rgt-out-- 58.7%

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

      7. mul-1-neg 58.7%

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

      8. mul-1-neg 58.7%

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

   7. Simplified 58.7%

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

      1. *-commutative 58.7%

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

      2. distribute-lft-out 58.7%

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

      3. add-sqr-sqrt 28.2%

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

      4. sqrt-unprod 61.8%

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

      5. sqr-neg 61.8%

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

      6. sqrt-unprod 33.6%

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

      7. add-sqr-sqrt 60.0%

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

      8. associate--l+ 60.0%

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

      9. metadata-eval 60.0%

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

   9. Applied egg-rr 60.0%

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

   10. Taylor expanded in x around 0 70.0%

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

       1. *-commutative 70.0%

          \[\leadsto \frac{2 + x \cdot \left(4 + \color{blue}{x \cdot 3}\right)}{2} \]

   12. Simplified 70.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(4 + x \cdot 3\right)}}{2} \]

   if 600 < x < 9.4999999999999998e77 or 9.9999999999999991e130 < x < 5.0000000000000002e205

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 69.5%

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

      1. mul-1-neg 69.5%

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

      2. mul-1-neg 69.5%

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

      3. rec-exp 69.5%

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

      4. sub-neg 69.5%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 69.5%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 69.5%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 69.5%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 69.5%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 69.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 9.4999999999999998e77 < x < 9.9999999999999991e130

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 16.7%

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

      1. associate--r+ 16.7%

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

      2. associate-*r* 16.7%

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

      3. mul-1-neg 16.7%

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

      4. cancel-sign-sub 16.7%

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

      5. distribute-rgt1-in 16.7%

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

      6. distribute-rgt-out-- 16.7%

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

      7. mul-1-neg 16.7%

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

      8. mul-1-neg 16.7%

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

   7. Simplified 16.7%

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

      1. *-commutative 16.7%

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

      2. distribute-lft-out 16.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 84.9%

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

      5. sqr-neg 84.9%

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

      6. sqrt-unprod 84.9%

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

      7. add-sqr-sqrt 84.9%

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

      8. associate--l+ 84.9%

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

      9. metadata-eval 84.9%

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

   9. Applied egg-rr 84.9%

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

   10. Taylor expanded in x around 0 56.2%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(4 + x \cdot \left(3 + 1.3333333333333333 \cdot x\right)\right)}}{2} \]
   11. Step-by-step derivation

       1. *-commutative 56.2%

          \[\leadsto \frac{2 + x \cdot \left(4 + x \cdot \left(3 + \color{blue}{x \cdot 1.3333333333333333}\right)\right)}{2} \]

   12. Simplified 56.2%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(4 + x \cdot \left(3 + x \cdot 1.3333333333333333\right)\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 600:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+131}:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot \left(3 + x \cdot 1.3333333333333333\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+205}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 540.0)
   (/ (+ 2.0 (* x (+ 4.0 (* x 3.0)))) 2.0)
   (if (<= x 1.55e+77)
     0.0
     (/
      (* x (+ 2.0 (* x (+ 2.0 (* x (+ 1.0 (* x 0.3333333333333333)))))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 540.0) {
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	} else if (x <= 1.55e+77) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 540.0d0) then
        tmp = (2.0d0 + (x * (4.0d0 + (x * 3.0d0)))) / 2.0d0
    else if (x <= 1.55d+77) then
        tmp = 0.0d0
    else
        tmp = (x * (2.0d0 + (x * (2.0d0 + (x * (1.0d0 + (x * 0.3333333333333333d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 540.0) {
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	} else if (x <= 1.55e+77) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 540.0:
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0
	elif x <= 1.55e+77:
		tmp = 0.0
	else:
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 540.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(4.0 + Float64(x * 3.0)))) / 2.0);
	elseif (x <= 1.55e+77)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.3333333333333333))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 540.0)
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	elseif (x <= 1.55e+77)
		tmp = 0.0;
	else
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 540.0], N[(N[(2.0 + N[(x * N[(4.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.55e+77], 0.0, N[(N[(x * N[(2.0 + N[(x * N[(2.0 + N[(x * N[(1.0 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 540

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in eps around 0 61.3%

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

      1. associate--r+ 61.3%

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

      2. associate-*r* 61.3%

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

      3. mul-1-neg 61.3%

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

      4. cancel-sign-sub 61.3%

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

      5. distribute-rgt1-in 61.3%

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

      6. distribute-rgt-out-- 61.9%

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

      7. mul-1-neg 61.9%

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

      8. mul-1-neg 61.9%

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

   7. Simplified 61.9%

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

      1. *-commutative 61.9%

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

      2. distribute-lft-out 61.9%

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

      3. add-sqr-sqrt 31.6%

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

      4. sqrt-unprod 60.9%

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

      5. sqr-neg 60.9%

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

      6. sqrt-unprod 29.3%

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

      7. add-sqr-sqrt 59.0%

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

      8. associate--l+ 59.0%

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

      9. metadata-eval 59.0%

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

   9. Applied egg-rr 59.0%

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

   10. Taylor expanded in x around 0 70.1%

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

       1. *-commutative 70.1%

          \[\leadsto \frac{2 + x \cdot \left(4 + \color{blue}{x \cdot 3}\right)}{2} \]

   12. Simplified 70.1%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(4 + x \cdot 3\right)}}{2} \]

   if 540 < x < 1.54999999999999999e77

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. mul-1-neg 71.3%

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

      3. rec-exp 71.3%

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

      4. sub-neg 71.3%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 71.3%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 71.3%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 71.3%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 71.3%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 71.3%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.54999999999999999e77 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 40.6%

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

      1. associate--r+ 40.6%

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

      2. associate-*r* 40.6%

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

      3. mul-1-neg 40.6%

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

      4. cancel-sign-sub 40.6%

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

      5. distribute-rgt1-in 40.6%

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

      6. distribute-rgt-out-- 40.6%

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

      7. mul-1-neg 40.6%

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

      8. mul-1-neg 40.6%

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

   7. Simplified 40.6%

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

      1. *-commutative 40.6%

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

      2. distribute-lft-out 40.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 61.0%

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

      5. sqr-neg 61.0%

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

      6. sqrt-unprod 61.0%

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

      7. add-sqr-sqrt 61.0%

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

      8. associate--l+ 61.0%

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

      9. metadata-eval 61.0%

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

   9. Applied egg-rr 61.0%

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

   10. Taylor expanded in x around inf 61.0%

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

       1. *-commutative 61.0%

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

       2. *-commutative 61.0%

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

       3. associate-*r* 61.0%

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

   12. Simplified 61.0%

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

   13. Taylor expanded in x around 0 61.0%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + 0.3333333333333333 \cdot x\right)\right)\right)}}{2} \]
   14. Step-by-step derivation

       1. *-commutative 61.0%

          \[\leadsto \frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + \color{blue}{x \cdot 0.3333333333333333}\right)\right)\right)}{2} \]

   15. Simplified 61.0%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{if}\;x \leq -550:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+206}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 2.0 (* x 2.0))) 2.0)))
   (if (<= x -550.0) t_0 (if (<= x 520.0) 1.0 (if (<= x 1.5e+206) 0.0 t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -550.0) {
		tmp = t_0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 1.5e+206) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (2.0d0 + (x * 2.0d0))) / 2.0d0
    if (x <= (-550.0d0)) then
        tmp = t_0
    else if (x <= 520.0d0) then
        tmp = 1.0d0
    else if (x <= 1.5d+206) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -550.0) {
		tmp = t_0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 1.5e+206) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x * (2.0 + (x * 2.0))) / 2.0
	tmp = 0
	if x <= -550.0:
		tmp = t_0
	elif x <= 520.0:
		tmp = 1.0
	elif x <= 1.5e+206:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x * Float64(2.0 + Float64(x * 2.0))) / 2.0)
	tmp = 0.0
	if (x <= -550.0)
		tmp = t_0;
	elseif (x <= 520.0)
		tmp = 1.0;
	elseif (x <= 1.5e+206)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	tmp = 0.0;
	if (x <= -550.0)
		tmp = t_0;
	elseif (x <= 520.0)
		tmp = 1.0;
	elseif (x <= 1.5e+206)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -550.0], t$95$0, If[LessEqual[x, 520.0], 1.0, If[LessEqual[x, 1.5e+206], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -550 or 1.5000000000000001e206 < x

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in eps around 0 13.2%

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

      1. associate--r+ 13.2%

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

      2. associate-*r* 13.2%

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

      3. mul-1-neg 13.2%

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

      4. cancel-sign-sub 13.2%

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

      5. distribute-rgt1-in 13.2%

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

      6. distribute-rgt-out-- 15.0%

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

      7. mul-1-neg 15.0%

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

      8. mul-1-neg 15.0%

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

   7. Simplified 15.0%

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

      1. *-commutative 15.0%

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

      2. distribute-lft-out 15.0%

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

      3. add-sqr-sqrt 1.8%

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

      4. sqrt-unprod 29.3%

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

      5. sqr-neg 29.3%

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

      6. sqrt-unprod 27.5%

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

      7. add-sqr-sqrt 28.4%

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

      8. associate--l+ 28.4%

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

      9. metadata-eval 28.4%

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

   9. Applied egg-rr 28.4%

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

   10. Taylor expanded in x around inf 28.4%

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

       1. *-commutative 28.4%

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

       2. *-commutative 28.4%

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

       3. associate-*r* 28.4%

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

   12. Simplified 28.4%

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

   13. Taylor expanded in x around 0 64.8%

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

       1. *-commutative 64.8%

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

   15. Simplified 64.8%

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

   if -550 < x < 520

   1. Initial program 52.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in x around 0 72.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 520 < x < 1.5000000000000001e206

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 57.0%

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

      1. mul-1-neg 57.0%

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

      2. mul-1-neg 57.0%

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

      3. rec-exp 57.0%

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

      4. sub-neg 57.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 57.0%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 57.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 57.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 57.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 57.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+206}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(eps\_m \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(eps\_m \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* (* eps_m x) -0.5)
   (if (<= x 480.0) 1.0 (if (<= x 2.1e+207) 0.0 (* (* eps_m x) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps_m * x) * -0.5;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 2.1e+207) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (eps_m * x) * (-0.5d0)
    else if (x <= 480.0d0) then
        tmp = 1.0d0
    else if (x <= 2.1d+207) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps_m * x) * -0.5;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 2.1e+207) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (eps_m * x) * -0.5
	elif x <= 480.0:
		tmp = 1.0
	elif x <= 2.1e+207:
		tmp = 0.0
	else:
		tmp = (eps_m * x) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(eps_m * x) * -0.5);
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 2.1e+207)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (eps_m * x) * -0.5;
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 2.1e+207)
		tmp = 0.0;
	else
		tmp = (eps_m * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(eps$95$m * x), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 2.1e+207], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 91.5%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in eps around inf 91.5%

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

   6. Taylor expanded in x around 0 38.7%

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

      1. mul-1-neg 38.7%

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

   8. Simplified 38.7%

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

   9. Taylor expanded in eps around inf 27.4%

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

       1. mul-1-neg 27.4%

          \[\leadsto \frac{\color{blue}{-\varepsilon \cdot x}}{2} \]

       2. *-commutative 27.4%

          \[\leadsto \frac{-\color{blue}{x \cdot \varepsilon}}{2} \]

       3. distribute-rgt-neg-in 27.4%

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

   11. Simplified 27.4%

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

       1. frac-2neg 27.4%

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

       2. div-inv 27.4%

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

       3. distribute-rgt-neg-out 27.4%

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

       4. remove-double-neg 27.4%

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

       5. metadata-eval 27.4%

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

       6. metadata-eval 27.4%

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

   13. Applied egg-rr 27.4%

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

   if -1 < x < 480

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in x around 0 73.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 480 < x < 2.0999999999999999e207

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 57.0%

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

      1. mul-1-neg 57.0%

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

      2. mul-1-neg 57.0%

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

      3. rec-exp 57.0%

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

      4. sub-neg 57.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 57.0%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 57.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 57.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 57.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 57.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 2.0999999999999999e207 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 23.6%

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

      1. mul-1-neg 23.6%

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

   8. Simplified 23.6%

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

   9. Taylor expanded in eps around inf 14.7%

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

       1. mul-1-neg 14.7%

          \[\leadsto \frac{\color{blue}{-\varepsilon \cdot x}}{2} \]

       2. *-commutative 14.7%

          \[\leadsto \frac{-\color{blue}{x \cdot \varepsilon}}{2} \]

       3. distribute-rgt-neg-in 14.7%

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

   11. Simplified 14.7%

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

       1. div-inv 14.7%

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

       2. add-sqr-sqrt 14.6%

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

       3. sqrt-unprod 69.7%

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

       4. sqr-neg 69.7%

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

       5. sqrt-unprod 41.9%

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

       6. add-sqr-sqrt 42.5%

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

       7. metadata-eval 42.5%

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

   13. Applied egg-rr 42.5%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.9% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(eps\_m \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (* (* eps_m x) -0.5) (if (<= x 600.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps_m * x) * -0.5;
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (eps_m * x) * (-0.5d0)
    else if (x <= 600.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps_m * x) * -0.5;
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (eps_m * x) * -0.5
	elif x <= 600.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(eps_m * x) * -0.5);
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (eps_m * x) * -0.5;
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(eps$95$m * x), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 600.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 91.5%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in eps around inf 91.5%

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

   6. Taylor expanded in x around 0 38.7%

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

      1. mul-1-neg 38.7%

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

   8. Simplified 38.7%

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

   9. Taylor expanded in eps around inf 27.4%

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

       1. mul-1-neg 27.4%

          \[\leadsto \frac{\color{blue}{-\varepsilon \cdot x}}{2} \]

       2. *-commutative 27.4%

          \[\leadsto \frac{-\color{blue}{x \cdot \varepsilon}}{2} \]

       3. distribute-rgt-neg-in 27.4%

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

   11. Simplified 27.4%

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

       1. frac-2neg 27.4%

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

       2. div-inv 27.4%

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

       3. distribute-rgt-neg-out 27.4%

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

       4. remove-double-neg 27.4%

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

       5. metadata-eval 27.4%

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

       6. metadata-eval 27.4%

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

   13. Applied egg-rr 27.4%

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

   if -1 < x < 600

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in x around 0 73.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 600 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 50.1%

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

      1. mul-1-neg 50.1%

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

      2. mul-1-neg 50.1%

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

      3. rec-exp 50.1%

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

      4. sub-neg 50.1%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 50.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 50.1%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 50.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 50.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 50.1%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.4% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 500.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 500.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 500.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in x around 0 59.6%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 50.1%

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

      1. mul-1-neg 50.1%

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

      2. mul-1-neg 50.1%

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

      3. rec-exp 50.1%

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

      4. sub-neg 50.1%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 50.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 50.1%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 50.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 50.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 50.1%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 16.2% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

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

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 72.4%

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

3. Simplified 61.5%

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

5. Taylor expanded in eps around 0 16.6%

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

   1. mul-1-neg 16.6%

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

   2. mul-1-neg 16.6%

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

   3. rec-exp 16.6%

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

   4. sub-neg 16.6%

      \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

   5. div-sub 16.6%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

   6. mul-1-neg 16.6%

      \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

   7. rec-exp 16.6%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

   8. +-inverses 16.8%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Simplified 16.8%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

8. Final simplification 16.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))