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

Percentage Accurate: 88.0% → 99.4%

Time: 6.9s

Alternatives: 8

Speedup: 2.0×

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: 88.0% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-319} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-319) (not (<= t_0 0.0)))
     t_0
     (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-319) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (eps * 5.0) * 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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-319)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-319) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -5e-319) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-319) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-319) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-319], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.9999937e-319 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 95.6%

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

   if -4.9999937e-319 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.5%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-319} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 2: 96.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-32}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.08e-32)
   (* (* eps 5.0) (pow x 4.0))
   (if (<= x 2.9e-38)
     (pow eps 5.0)
     (* eps (+ (* 5.0 (* (* x x) (* x x))) (* eps (* (pow x 3.0) 10.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.08e-32) {
		tmp = (eps * 5.0) * pow(x, 4.0);
	} else if (x <= 2.9e-38) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * ((5.0 * ((x * x) * (x * x))) + (eps * (pow(x, 3.0) * 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.08d-32)) then
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    else if (x <= 2.9d-38) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * ((5.0d0 * ((x * x) * (x * x))) + (eps * ((x ** 3.0d0) * 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.08e-32) {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	} else if (x <= 2.9e-38) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * ((5.0 * ((x * x) * (x * x))) + (eps * (Math.pow(x, 3.0) * 10.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.08e-32:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	elif x <= 2.9e-38:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * ((5.0 * ((x * x) * (x * x))) + (eps * (math.pow(x, 3.0) * 10.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.08e-32)
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	elseif (x <= 2.9e-38)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))) + Float64(eps * Float64((x ^ 3.0) * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.08e-32)
		tmp = (eps * 5.0) * (x ^ 4.0);
	elseif (x <= 2.9e-38)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((5.0 * ((x * x) * (x * x))) + (eps * ((x ^ 3.0) * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.08e-32], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-38], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08e-32

   1. Initial program 22.1%

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

   2. Taylor expanded in x around inf 91.3%

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

      1. distribute-lft1-in 91.3%

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

      2. metadata-eval 91.3%

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

      3. *-commutative 91.3%

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

   4. Simplified 91.3%

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

   if -1.08e-32 < x < 2.89999999999999994e-38

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 98.8%

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

   if 2.89999999999999994e-38 < x

   1. Initial program 29.8%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. associate-*l* 99.7%

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

      4. distribute-lft-out 99.6%

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

      5. distribute-lft1-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. *-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. distribute-rgt-out 99.6%

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

      10. associate-*r* 99.6%

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

   4. Simplified 99.6%

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

      1. metadata-eval 96.7%

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

      2. metadata-eval 96.7%

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

      3. metadata-eval 96.7%

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

      4. pow-prod-up 96.4%

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

      5. pow-prod-down 96.4%

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

      6. metadata-eval 96.4%

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

      7. pow2 96.4%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-32}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \]

Alternative 3: 96.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-32}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.08e-32)
   (* (* eps 5.0) (pow x 4.0))
   (if (<= x 3.5e-38) (pow eps 5.0) (* (* eps 5.0) (* x (pow x 3.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.08e-32) {
		tmp = (eps * 5.0) * pow(x, 4.0);
	} else if (x <= 3.5e-38) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (eps * 5.0) * (x * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.08d-32)) then
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    else if (x <= 3.5d-38) then
        tmp = eps ** 5.0d0
    else
        tmp = (eps * 5.0d0) * (x * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.08e-32) {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	} else if (x <= 3.5e-38) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (eps * 5.0) * (x * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.08e-32:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	elif x <= 3.5e-38:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (eps * 5.0) * (x * math.pow(x, 3.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.08e-32)
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	elseif (x <= 3.5e-38)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(eps * 5.0) * Float64(x * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.08e-32)
		tmp = (eps * 5.0) * (x ^ 4.0);
	elseif (x <= 3.5e-38)
		tmp = eps ^ 5.0;
	else
		tmp = (eps * 5.0) * (x * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.08e-32], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e-38], N[Power[eps, 5.0], $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08e-32

   1. Initial program 22.1%

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

   2. Taylor expanded in x around inf 91.3%

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

      1. distribute-lft1-in 91.3%

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

      2. metadata-eval 91.3%

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

      3. *-commutative 91.3%

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

   4. Simplified 91.3%

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

   if -1.08e-32 < x < 3.5000000000000001e-38

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 98.8%

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

   if 3.5000000000000001e-38 < x

   1. Initial program 29.8%

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

   2. Taylor expanded in x around inf 96.9%

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

      1. distribute-lft1-in 96.9%

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

      2. metadata-eval 96.9%

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

      3. *-commutative 96.9%

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

   4. Simplified 96.9%

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

      1. add-sqr-sqrt 95.4%

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

      2. unpow-prod-down 95.8%

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

   6. Applied egg-rr 95.8%

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

      1. pow-sqr 95.8%

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

      2. metadata-eval 95.8%

         \[\leadsto \left(\varepsilon \cdot 5\right) \cdot {\left(\sqrt{x}\right)}^{\color{blue}{8}} \]

   8. Simplified 95.8%

      \[\leadsto \left(\varepsilon \cdot 5\right) \cdot \color{blue}{{\left(\sqrt{x}\right)}^{8}} \]
   9. Step-by-step derivation

      1. sqrt-pow2 96.9%

         \[\leadsto \left(\varepsilon \cdot 5\right) \cdot \color{blue}{{x}^{\left(\frac{8}{2}\right)}} \]

      2. metadata-eval 96.9%

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

      3. metadata-eval 96.9%

         \[\leadsto \left(\varepsilon \cdot 5\right) \cdot {x}^{\color{blue}{\left(3 + 1\right)}} \]

      4. metadata-eval 96.9%

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

      5. metadata-eval 96.9%

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

      6. pow-prod-up 96.9%

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

      7. metadata-eval 96.9%

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

      8. metadata-eval 96.9%

         \[\leadsto \left(\varepsilon \cdot 5\right) \cdot \left({x}^{\color{blue}{3}} \cdot {x}^{1}\right) \]

      9. pow1 96.9%

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

   10. Applied egg-rr 96.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-32}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot {x}^{3}\right)\\ \end{array} \]

Alternative 4: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-32} \lor \neg \left(x \leq 2.5 \cdot 10^{-38}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.08e-32) (not (<= x 2.5e-38)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.08e-32) || !(x <= 2.5e-38)) {
		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 <= (-1.08d-32)) .or. (.not. (x <= 2.5d-38))) 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 <= -1.08e-32) || !(x <= 2.5e-38)) {
		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 <= -1.08e-32) or not (x <= 2.5e-38):
		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 <= -1.08e-32) || !(x <= 2.5e-38))
		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 <= -1.08e-32) || ~((x <= 2.5e-38)))
		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, -1.08e-32], N[Not[LessEqual[x, 2.5e-38]], $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 < -1.08e-32 or 2.50000000000000017e-38 < x

   1. Initial program 25.2%

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

   2. Taylor expanded in x around inf 93.5%

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

      1. distribute-lft1-in 93.5%

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

      2. metadata-eval 93.5%

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

      3. associate-*l* 93.4%

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

   4. Simplified 93.4%

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

   if -1.08e-32 < x < 2.50000000000000017e-38

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-32} \lor \neg \left(x \leq 2.5 \cdot 10^{-38}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-32} \lor \neg \left(x \leq 2.5 \cdot 10^{-38}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.12e-32) (not (<= x 2.5e-38)))
   (* (* eps 5.0) (pow x 4.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.12e-32) || !(x <= 2.5e-38)) {
		tmp = (eps * 5.0) * 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 <= (-1.12d-32)) .or. (.not. (x <= 2.5d-38))) then
        tmp = (eps * 5.0d0) * (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 <= -1.12e-32) || !(x <= 2.5e-38)) {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.12e-32) or not (x <= 2.5e-38):
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.12e-32) || !(x <= 2.5e-38))
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.12e-32) || ~((x <= 2.5e-38)))
		tmp = (eps * 5.0) * (x ^ 4.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.12e-32], N[Not[LessEqual[x, 2.5e-38]], $MachinePrecision]], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.12e-32 or 2.50000000000000017e-38 < x

   1. Initial program 25.2%

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

   2. Taylor expanded in x around inf 93.5%

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

      1. distribute-lft1-in 93.5%

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

      2. metadata-eval 93.5%

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

      3. *-commutative 93.5%

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

   4. Simplified 93.5%

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

   if -1.12e-32 < x < 2.50000000000000017e-38

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-32} \lor \neg \left(x \leq 2.5 \cdot 10^{-38}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6: 96.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{-32}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x -1.08e-32)
     (* (* eps 5.0) t_0)
     (if (<= x 2.5e-38) (pow eps 5.0) (* 5.0 (* eps t_0))))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -1.08e-32) {
		tmp = (eps * 5.0) * t_0;
	} else if (x <= 2.5e-38) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (x * x)
    if (x <= (-1.08d-32)) then
        tmp = (eps * 5.0d0) * t_0
    else if (x <= 2.5d-38) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -1.08e-32) {
		tmp = (eps * 5.0) * t_0;
	} else if (x <= 2.5e-38) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (x * x) * (x * x)
	tmp = 0
	if x <= -1.08e-32:
		tmp = (eps * 5.0) * t_0
	elif x <= 2.5e-38:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -1.08e-32)
		tmp = Float64(Float64(eps * 5.0) * t_0);
	elseif (x <= 2.5e-38)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x * x) * (x * x);
	tmp = 0.0;
	if (x <= -1.08e-32)
		tmp = (eps * 5.0) * t_0;
	elseif (x <= 2.5e-38)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.08e-32], N[(N[(eps * 5.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 2.5e-38], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08e-32

   1. Initial program 22.1%

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

   2. Taylor expanded in x around inf 91.3%

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

      1. distribute-lft1-in 91.3%

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

      2. metadata-eval 91.3%

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

      3. *-commutative 91.3%

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

   4. Simplified 91.3%

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

      1. metadata-eval 91.1%

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

      2. metadata-eval 91.1%

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

      3. metadata-eval 91.1%

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

      4. pow-prod-up 90.8%

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

      5. pow-prod-down 90.8%

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

      6. metadata-eval 90.8%

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

      7. pow2 90.8%

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

   6. Applied egg-rr 90.9%

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

   if -1.08e-32 < x < 2.50000000000000017e-38

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 98.8%

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

   if 2.50000000000000017e-38 < x

   1. Initial program 29.8%

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

   2. Taylor expanded in x around inf 96.9%

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

      1. distribute-lft1-in 96.9%

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

      2. metadata-eval 96.9%

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

      3. associate-*l* 96.7%

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

   4. Simplified 96.7%

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

      1. metadata-eval 96.7%

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

      2. metadata-eval 96.7%

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

      3. metadata-eval 96.7%

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

      4. pow-prod-up 96.4%

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

      5. pow-prod-down 96.4%

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

      6. metadata-eval 96.4%

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

      7. pow2 96.4%

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

   6. Applied egg-rr 96.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-32}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 7: 82.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 5.0 * (eps * ((x * x) * (x * x)));
}

Python

def code(x, eps):
	return 5.0 * (eps * ((x * x) * (x * x)))

Julia

function code(x, eps)
	return Float64(5.0 * Float64(eps * Float64(Float64(x * x) * Float64(x * x))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around inf 82.1%

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

   1. distribute-lft1-in 82.1%

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

   2. metadata-eval 82.1%

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

   3. associate-*l* 82.0%

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

4. Simplified 82.0%

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

   1. metadata-eval 82.0%

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

   2. metadata-eval 82.0%

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

   3. metadata-eval 82.0%

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

   4. pow-prod-up 82.0%

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

   5. pow-prod-down 82.0%

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

   6. metadata-eval 82.0%

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

   7. pow2 82.0%

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

6. Applied egg-rr 82.0%

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

7. Final simplification 82.0%

   \[\leadsto 5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

Alternative 8: 82.5% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return (eps * 5.0) * ((x * x) * (x * x));
}

Python

def code(x, eps):
	return (eps * 5.0) * ((x * x) * (x * x))

Julia

function code(x, eps)
	return Float64(Float64(eps * 5.0) * Float64(Float64(x * x) * Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around inf 82.1%

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

   1. distribute-lft1-in 82.1%

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

   2. metadata-eval 82.1%

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

   3. *-commutative 82.1%

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

4. Simplified 82.1%

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

   1. metadata-eval 82.0%

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

   2. metadata-eval 82.0%

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

   3. metadata-eval 82.0%

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

   4. pow-prod-up 82.0%

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

   5. pow-prod-down 82.0%

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

   6. metadata-eval 82.0%

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

   7. pow2 82.0%

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

6. Applied egg-rr 82.0%

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

7. Final simplification 82.0%

   \[\leadsto \left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

Reproduce

herbie shell --seed 2023238 
(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)))