ENA, Section 1.4, Exercise 4b, n=5

Percentage Accurate: 88.2% → 97.9%

Time: 9.5s

Alternatives: 12

Speedup: 1.8×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot {\varepsilon}^{3}}{x} - {\varepsilon}^{2} \cdot -10}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* (pow x 3.0) (+ (* x 5.0) (* eps 10.0))))
   (if (<= x 5.5e-39)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (*
      (pow x 4.0)
      (+
       (* eps 5.0)
       (/ (- (/ (* 10.0 (pow eps 3.0)) x) (* (pow eps 2.0) -10.0)) x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 5.5e-39) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * pow(eps, 3.0)) / x) - (pow(eps, 2.0) * -10.0)) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.2d-43)) then
        tmp = eps * ((x ** 3.0d0) * ((x * 5.0d0) + (eps * 10.0d0)))
    else if (x <= 5.5d-39) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) + ((((10.0d0 * (eps ** 3.0d0)) / x) - ((eps ** 2.0d0) * (-10.0d0))) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (Math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 5.5e-39) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * Math.pow(eps, 3.0)) / x) - (Math.pow(eps, 2.0) * -10.0)) / x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-43:
		tmp = eps * (math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)))
	elif x <= 5.5e-39:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * math.pow(eps, 3.0)) / x) - (math.pow(eps, 2.0) * -10.0)) / x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64((x ^ 3.0) * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	elseif (x <= 5.5e-39)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) + Float64(Float64(Float64(Float64(10.0 * (eps ^ 3.0)) / x) - Float64((eps ^ 2.0) * -10.0)) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-43)
		tmp = eps * ((x ^ 3.0) * ((x * 5.0) + (eps * 10.0)));
	elseif (x <= 5.5e-39)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = (x ^ 4.0) * ((eps * 5.0) + ((((10.0 * (eps ^ 3.0)) / x) - ((eps ^ 2.0) * -10.0)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-39], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(N[(N[(N[(10.0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[Power[eps, 2.0], $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 32.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+r+ 99.7%

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

      3. distribute-lft1-in 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. fma-define 99.8%

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* 99.8%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{3}}\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \left(10 \cdot \varepsilon\right) \cdot {x}^{3}\right)} \]

   9. Taylor expanded in x around 0 99.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

   if -3.19999999999999985e-43 < x < 5.50000000000000018e-39

   1. Initial program 99.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   if 5.50000000000000018e-39 < x

   1. Initial program 22.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10 - \frac{{\varepsilon}^{3} \cdot 10}{x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot {\varepsilon}^{3}}{x} - {\varepsilon}^{2} \cdot -10}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.9e-43)
   (* eps (* (pow x 3.0) (+ (* x 5.0) (* eps 10.0))))
   (if (<= x 5.5e-39)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* (pow x 3.0) (+ (* 5.0 (* x eps)) (* 10.0 (pow eps 2.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.9e-43) {
		tmp = eps * (pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 5.5e-39) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * pow(eps, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.9d-43)) then
        tmp = eps * ((x ** 3.0d0) * ((x * 5.0d0) + (eps * 10.0d0)))
    else if (x <= 5.5d-39) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = (x ** 3.0d0) * ((5.0d0 * (x * eps)) + (10.0d0 * (eps ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.9e-43) {
		tmp = eps * (Math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 5.5e-39) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = Math.pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * Math.pow(eps, 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.9e-43:
		tmp = eps * (math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)))
	elif x <= 5.5e-39:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = math.pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * math.pow(eps, 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.9e-43)
		tmp = Float64(eps * Float64((x ^ 3.0) * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	elseif (x <= 5.5e-39)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64((x ^ 3.0) * Float64(Float64(5.0 * Float64(x * eps)) + Float64(10.0 * (eps ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.9e-43)
		tmp = eps * ((x ^ 3.0) * ((x * 5.0) + (eps * 10.0)));
	elseif (x <= 5.5e-39)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = (x ^ 3.0) * ((5.0 * (x * eps)) + (10.0 * (eps ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.9e-43], N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-39], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(5.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.89999999999999985e-43

   1. Initial program 32.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+r+ 99.7%

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

      3. distribute-lft1-in 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. fma-define 99.8%

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* 99.8%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{3}}\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \left(10 \cdot \varepsilon\right) \cdot {x}^{3}\right)} \]

   9. Taylor expanded in x around 0 99.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

   if -1.89999999999999985e-43 < x < 5.50000000000000018e-39

   1. Initial program 99.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   if 5.50000000000000018e-39 < x

   1. Initial program 22.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r+ 99.6%

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

      3. distribute-lft1-in 99.6%

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

      4. metadata-eval 99.6%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 99.6%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 99.6%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 99.6%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.1e-41)
   (* eps (* (pow x 3.0) (+ (* x 5.0) (* eps 10.0))))
   (if (<= x 6e-39)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-41) {
		tmp = eps * (pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 6e-39) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.1d-41)) then
        tmp = eps * ((x ** 3.0d0) * ((x * 5.0d0) + (eps * 10.0d0)))
    else if (x <= 6d-39) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = eps * ((x ** 4.0d0) * (5.0d0 + (10.0d0 * (eps / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-41) {
		tmp = eps * (Math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 6e-39) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = eps * (Math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.1e-41:
		tmp = eps * (math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)))
	elif x <= 6e-39:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = eps * (math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.1e-41)
		tmp = Float64(eps * Float64((x ^ 3.0) * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	elseif (x <= 6e-39)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.1e-41)
		tmp = eps * ((x ^ 3.0) * ((x * 5.0) + (eps * 10.0)));
	elseif (x <= 6e-39)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = eps * ((x ^ 4.0) * (5.0 + (10.0 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.1e-41], N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-39], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.10000000000000013e-41

   1. Initial program 32.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+r+ 99.7%

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

      3. distribute-lft1-in 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 99.7%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. fma-define 99.8%

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* 99.8%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{3}}\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \left(10 \cdot \varepsilon\right) \cdot {x}^{3}\right)} \]

   9. Taylor expanded in x around 0 99.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

   if -2.10000000000000013e-41 < x < 6.00000000000000055e-39

   1. Initial program 99.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   if 6.00000000000000055e-39 < x

   1. Initial program 22.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r+ 99.6%

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

      3. distribute-lft1-in 99.6%

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

      4. metadata-eval 99.6%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 99.6%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 99.6%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 99.6%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 99.6%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. fma-define 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* 99.5%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{3}}\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \left(10 \cdot \varepsilon\right) \cdot {x}^{3}\right)} \]

   9. Taylor expanded in x around inf 99.6%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-52} \lor \neg \left(x \leq 7.2 \cdot 10^{-46}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-52) (not (<= x 7.2e-46)))
   (* eps (* (pow x 3.0) (+ (* x 5.0) (* eps 10.0))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-52) || !(x <= 7.2e-46)) {
		tmp = eps * (pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5d-52)) .or. (.not. (x <= 7.2d-46))) then
        tmp = eps * ((x ** 3.0d0) * ((x * 5.0d0) + (eps * 10.0d0)))
    else
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-52) || !(x <= 7.2e-46)) {
		tmp = eps * (Math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5e-52) or not (x <= 7.2e-46):
		tmp = eps * (math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)))
	else:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-52) || !(x <= 7.2e-46))
		tmp = Float64(eps * Float64((x ^ 3.0) * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	else
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5e-52) || ~((x <= 7.2e-46)))
		tmp = eps * ((x ^ 3.0) * ((x * 5.0) + (eps * 10.0)));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-52], N[Not[LessEqual[x, 7.2e-46]], $MachinePrecision]], N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e-52 or 7.2e-46 < x

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 95.1%

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

      1. +-commutative 95.1%

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

      2. associate-+r+ 95.1%

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

      3. distribute-lft1-in 95.1%

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

      4. metadata-eval 95.1%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 95.1%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 95.1%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 95.1%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 95.1%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 95.1%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 95.1%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 95.1%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 95.1%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in eps around 0 95.1%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. fma-define 95.1%

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* 95.1%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{3}}\right) \]

   8. Simplified 95.1%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \left(10 \cdot \varepsilon\right) \cdot {x}^{3}\right)} \]

   9. Taylor expanded in x around 0 95.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

   if -5e-52 < x < 7.2e-46

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft1-in 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}}\right) \]

      2. metadata-eval 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{5} \cdot \frac{x}{\varepsilon}\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-52} \lor \neg \left(x \leq 7.2 \cdot 10^{-46}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.6e-51)
   (* eps (* (pow x 3.0) (+ (* x 5.0) (* eps 10.0))))
   (if (<= x 2.8e-46)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.6e-51) {
		tmp = eps * (pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 2.8e-46) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.6d-51)) then
        tmp = eps * ((x ** 3.0d0) * ((x * 5.0d0) + (eps * 10.0d0)))
    else if (x <= 2.8d-46) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * ((x ** 4.0d0) * (5.0d0 + (10.0d0 * (eps / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.6e-51) {
		tmp = eps * (Math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 2.8e-46) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (Math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.6e-51:
		tmp = eps * (math.pow(x, 3.0) * ((x * 5.0) + (eps * 10.0)))
	elif x <= 2.8e-46:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * (math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.6e-51)
		tmp = Float64(eps * Float64((x ^ 3.0) * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	elseif (x <= 2.8e-46)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.6e-51)
		tmp = eps * ((x ^ 3.0) * ((x * 5.0) + (eps * 10.0)));
	elseif (x <= 2.8e-46)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * ((x ^ 4.0) * (5.0 + (10.0 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.6e-51], N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-46], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e-51

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 96.8%

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

      1. +-commutative 96.8%

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

      2. associate-+r+ 96.8%

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

      3. distribute-lft1-in 96.8%

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

      4. metadata-eval 96.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 96.8%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 96.8%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 96.8%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 96.8%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 96.8%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 96.8%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 96.8%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 96.8%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 96.8%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in eps around 0 96.8%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. fma-define 96.8%

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* 96.8%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{3}}\right) \]

   8. Simplified 96.8%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \left(10 \cdot \varepsilon\right) \cdot {x}^{3}\right)} \]

   9. Taylor expanded in x around 0 96.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

   if -1.6e-51 < x < 2.7999999999999998e-46

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft1-in 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}}\right) \]

      2. metadata-eval 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{5} \cdot \frac{x}{\varepsilon}\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]

   if 2.7999999999999998e-46 < x

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. associate-+r+ 93.9%

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

      3. distribute-lft1-in 93.9%

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

      4. metadata-eval 93.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4} + \varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      5. *-commutative 93.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{{x}^{3} \cdot 4} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) \]

      6. distribute-rgt-out 93.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right)\right) \]

      7. associate-*r* 93.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(2 + 4\right)}\right)\right) \]

      8. unpow2 93.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(2 + 4\right)\right)\right) \]

      9. cube-mult 93.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 4 + \color{blue}{{x}^{3}} \cdot \left(2 + 4\right)\right)\right) \]

      10. distribute-lft-out 93.9%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{3} \cdot \left(4 + \left(2 + 4\right)\right)\right)}\right) \]

      11. metadata-eval 93.9%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 93.9%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \color{blue}{10}\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]

   6. Taylor expanded in eps around 0 93.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. fma-define 93.9%

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* 93.9%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{3}}\right) \]

   8. Simplified 93.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \left(10 \cdot \varepsilon\right) \cdot {x}^{3}\right)} \]

   9. Taylor expanded in x around inf 93.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.4e-47)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 9.2e-48)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-47) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 9.2e-48) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.4d-47)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 9.2d-48) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-47) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 9.2e-48) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.4e-47:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 9.2e-48:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.4e-47)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 9.2e-48)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.4e-47)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 9.2e-48)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.4e-47], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e-48], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999996e-47

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.5%

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

      1. *-commutative 94.5%

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

      2. distribute-rgt1-in 94.5%

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

      3. metadata-eval 94.5%

         \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      4. *-commutative 94.5%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

      5. associate-*r* 94.6%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   if -1.39999999999999996e-47 < x < 9.2000000000000003e-48

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft1-in 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}}\right) \]

      2. metadata-eval 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{5} \cdot \frac{x}{\varepsilon}\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]

   if 9.2000000000000003e-48 < x

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.9%

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

      1. distribute-rgt1-in 92.9%

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

      2. metadata-eval 92.9%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(5 \cdot \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.1e-47)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 3.5e-49)
     (* (pow eps 4.0) (+ eps (* x 5.0)))
     (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-47) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 3.5e-49) {
		tmp = pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.1d-47)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 3.5d-49) then
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-47) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 3.5e-49) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.1e-47:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 3.5e-49:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.1e-47)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 3.5e-49)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.1e-47)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 3.5e-49)
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.1e-47], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e-49], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1000000000000001e-47

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.5%

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

      1. *-commutative 94.5%

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

      2. distribute-rgt1-in 94.5%

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

      3. metadata-eval 94.5%

         \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      4. *-commutative 94.5%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

      5. associate-*r* 94.6%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   if -2.1000000000000001e-47 < x < 3.50000000000000006e-49

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft1-in 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}}\right) \]

      2. metadata-eval 99.7%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{5} \cdot \frac{x}{\varepsilon}\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]

   6. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]

   if 3.50000000000000006e-49 < x

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.9%

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

      1. distribute-rgt1-in 92.9%

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

      2. metadata-eval 92.9%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(5 \cdot \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50} \lor \neg \left(x \leq 4 \cdot 10^{-49}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.5e-50) (not (<= x 4e-49)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.5e-50) || !(x <= 4e-49)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.5d-50)) .or. (.not. (x <= 4d-49))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.5e-50) || !(x <= 4e-49)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.5e-50) or not (x <= 4e-49):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.5e-50) || !(x <= 4e-49))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.5e-50) || ~((x <= 4e-49)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.5e-50], N[Not[LessEqual[x, 4e-49]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999995e-50 or 3.99999999999999975e-49 < x

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 35.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \cdot \sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}\right) \cdot \sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}} \]

      2. pow3 35.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}\right)}^{3}} \]

   4. Applied egg-rr 35.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}\right)}^{3}} \]

   5. Taylor expanded in x around inf 93.5%

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

      1. distribute-rgt1-in 93.5%

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

      2. metadata-eval 93.5%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      3. *-commutative 93.5%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\varepsilon \cdot 5\right)} \]

      4. *-commutative 93.5%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \]

      5. *-commutative 93.5%

         \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right)} \cdot {x}^{4} \]

      6. associate-*r* 93.4%

         \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

   7. Simplified 93.4%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

   if -1.49999999999999995e-50 < x < 3.99999999999999975e-49

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50} \lor \neg \left(x \leq 4 \cdot 10^{-49}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.6e-52)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 6.2e-49) (pow eps 5.0) (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-52) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 6.2e-49) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.6d-52)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 6.2d-49) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-52) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 6.2e-49) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.6e-52:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 6.2e-49:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.6e-52)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 6.2e-49)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.6e-52)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 6.2e-49)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.6e-52], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e-49], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5999999999999999e-52

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.5%

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

      1. *-commutative 94.5%

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

      2. distribute-rgt1-in 94.5%

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

      3. metadata-eval 94.5%

         \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      4. *-commutative 94.5%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

      5. associate-*r* 94.6%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   if -6.5999999999999999e-52 < x < 6.2e-49

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 6.2e-49 < x

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.9%

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

      1. distribute-rgt1-in 92.9%

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

      2. metadata-eval 92.9%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(5 \cdot \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-50}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-50)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1.6e-49) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-50) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1.6e-49) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-9d-50)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 1.6d-49) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9e-50) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 1.6e-49) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9e-50:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 1.6e-49:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-50)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1.6e-49)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9e-50)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 1.6e-49)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-50], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-49], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999924e-50

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.5%

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

      1. *-commutative 94.5%

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

      2. distribute-rgt1-in 94.5%

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

      3. metadata-eval 94.5%

         \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      4. *-commutative 94.5%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

      5. associate-*r* 94.6%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   if -8.99999999999999924e-50 < x < 1.60000000000000001e-49

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 1.60000000000000001e-49 < x

   1. Initial program 35.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 35.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \cdot \sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}\right) \cdot \sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}} \]

      2. pow3 35.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}\right)}^{3}} \]

   4. Applied egg-rr 35.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + \varepsilon\right)}^{5} - {x}^{5}}\right)}^{3}} \]

   5. Taylor expanded in x around inf 92.9%

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

      1. distribute-rgt1-in 92.9%

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

      2. metadata-eval 92.9%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      3. *-commutative 92.9%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\varepsilon \cdot 5\right)} \]

      4. *-commutative 92.9%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \]

      5. *-commutative 92.9%

         \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right)} \cdot {x}^{4} \]

      6. associate-*r* 92.9%

         \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

   7. Simplified 92.9%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (pow eps 5.0))

C

double code(double x, double eps) {
	return pow(eps, 5.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps ** 5.0d0
end function

Java

public static double code(double x, double eps) {
	return Math.pow(eps, 5.0);
}

Python

def code(x, eps):
	return math.pow(eps, 5.0)

Julia

function code(x, eps)
	return eps ^ 5.0
end

MATLAB

function tmp = code(x, eps)
	tmp = eps ^ 5.0;
end

Wolfram

code[x_, eps_] := N[Power[eps, 5.0], $MachinePrecision]

Derivation

1. Initial program 86.6%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 85.0%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
4. Add Preprocessing

Alternative 12: 71.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 86.6%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 86.6%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} + \left(-{x}^{5}\right)} \]

   2. +-commutative 86.6%

      \[\leadsto \color{blue}{\left(-{x}^{5}\right) + {\left(x + \varepsilon\right)}^{5}} \]

   3. add-sqr-sqrt 78.3%

      \[\leadsto \color{blue}{\sqrt{-{x}^{5}} \cdot \sqrt{-{x}^{5}}} + {\left(x + \varepsilon\right)}^{5} \]

   4. fma-define 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-{x}^{5}}, \sqrt{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right)} \]

4. Applied egg-rr 76.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-{x}^{5}}, \sqrt{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right)} \]

5. Taylor expanded in x around inf 0.0%

   \[\leadsto \color{blue}{{x}^{5} \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \color{blue}{\left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot {x}^{5}} \]

   2. unpow2 0.0%

      \[\leadsto \left(1 + \color{blue}{\sqrt{-1} \cdot \sqrt{-1}}\right) \cdot {x}^{5} \]

   3. rem-square-sqrt 71.8%

      \[\leadsto \left(1 + \color{blue}{-1}\right) \cdot {x}^{5} \]

   4. metadata-eval 71.8%

      \[\leadsto \color{blue}{0} \cdot {x}^{5} \]

   5. mul0-lft 71.8%

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

7. Simplified 71.8%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))