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

Percentage Accurate: 88.8% → 99.0%

Time: 10.6s

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: 88.8% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-296} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-296) (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-296) || !(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-296)) .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-296) || !(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-296) 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-296) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(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-296) || ~((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-296], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -5.0000000000000003e-296 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.7%

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

   if -5.0000000000000003e-296 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.1%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-296} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 2: 96.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-56} \lor \neg \left(x \leq 2.75 \cdot 10^{-90}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -9e-56) (not (<= x 2.75e-90)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -9e-56) || !(x <= 2.75e-90)) {
		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 <= (-9d-56)) .or. (.not. (x <= 2.75d-90))) 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 <= -9e-56) || !(x <= 2.75e-90)) {
		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 <= -9e-56) or not (x <= 2.75e-90):
		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 <= -9e-56) || !(x <= 2.75e-90))
		tmp = Float64(eps * Float64(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 <= -9e-56) || ~((x <= 2.75e-90)))
		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, -9e-56], N[Not[LessEqual[x, 2.75e-90]], $MachinePrecision]], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.0000000000000001e-56 or 2.75000000000000015e-90 < x

   1. Initial program 49.4%

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

   2. Taylor expanded in x around inf 95.8%

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

      1. *-commutative 95.8%

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

      2. distribute-rgt1-in 95.8%

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

      3. metadata-eval 95.8%

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

      4. *-commutative 95.8%

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

      5. associate-*r* 95.8%

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

   4. Simplified 95.8%

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

   if -9.0000000000000001e-56 < x < 2.75000000000000015e-90

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-56} \lor \neg \left(x \leq 2.75 \cdot 10^{-90}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 3: 96.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-90}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.5e-56)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 2.75e-90) (pow eps 5.0) (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.5e-56) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 2.75e-90) {
		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 <= (-7.5d-56)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 2.75d-90) 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 <= -7.5e-56) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 2.75e-90) {
		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 <= -7.5e-56:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 2.75e-90:
		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 <= -7.5e-56)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 2.75e-90)
		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 <= -7.5e-56)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 2.75e-90)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.5e-56], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.75e-90], 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 < -7.50000000000000041e-56

   1. Initial program 33.3%

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

   2. Taylor expanded in x around inf 97.1%

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

      1. *-commutative 97.1%

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

      2. distribute-rgt1-in 97.1%

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

      3. metadata-eval 97.1%

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

      4. *-commutative 97.1%

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

      5. associate-*r* 97.4%

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

   4. Simplified 97.4%

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

   if -7.50000000000000041e-56 < x < 2.75000000000000015e-90

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.3%

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

   if 2.75000000000000015e-90 < x

   1. Initial program 57.4%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. distribute-rgt1-in 95.1%

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

      2. metadata-eval 95.1%

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

   4. Simplified 95.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-90}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-90}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.8e-56)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 2.75e-90) (pow eps 5.0) (* (* x x) (* (* eps 5.0) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.8e-56) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 2.75e-90) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.8d-56)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 2.75d-90) then
        tmp = eps ** 5.0d0
    else
        tmp = (x * x) * ((eps * 5.0d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.8e-56) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 2.75e-90) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.8e-56:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 2.75e-90:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (x * x) * ((eps * 5.0) * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.8e-56)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 2.75e-90)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * 5.0) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.8e-56)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 2.75e-90)
		tmp = eps ^ 5.0;
	else
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.8e-56], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.75e-90], N[Power[eps, 5.0], $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999964e-56

   1. Initial program 33.3%

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

   2. Taylor expanded in x around inf 97.1%

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

      1. distribute-rgt1-in 97.1%

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

      2. metadata-eval 97.1%

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

   4. Simplified 97.1%

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

   5. Taylor expanded in x around 0 97.2%

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

   if -6.79999999999999964e-56 < x < 2.75000000000000015e-90

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.3%

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

   if 2.75000000000000015e-90 < x

   1. Initial program 57.4%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. distribute-rgt1-in 95.1%

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

      2. metadata-eval 95.1%

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

   4. Simplified 95.1%

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

      1. add-sqr-sqrt 67.7%

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

      2. sqrt-unprod 57.3%

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

      3. associate-*r* 57.3%

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

      4. *-commutative 57.3%

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

      5. associate-*r* 57.3%

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

      6. *-commutative 57.3%

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

      7. swap-sqr 57.3%

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

      8. *-commutative 57.3%

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

      9. *-commutative 57.3%

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

      10. swap-sqr 57.3%

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

      11. pow-prod-up 57.3%

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

      12. metadata-eval 57.3%

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

      13. metadata-eval 57.3%

          \[\leadsto \sqrt{\left({x}^{8} \cdot \color{blue}{25}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   6. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\sqrt{\left({x}^{8} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}} \]
   7. Step-by-step derivation

      1. metadata-eval 57.3%

         \[\leadsto \sqrt{\left({x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      2. pow-sqr 57.3%

         \[\leadsto \sqrt{\left(\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      3. associate-*l* 57.3%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]

      4. pow-sqr 57.3%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      5. metadata-eval 57.3%

         \[\leadsto \sqrt{{x}^{\color{blue}{8}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   8. Simplified 57.3%

      \[\leadsto \color{blue}{\sqrt{{x}^{8} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]
   9. Step-by-step derivation

      1. metadata-eval 57.3%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      2. pow-sqr 57.3%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      3. metadata-eval 57.3%

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

      4. swap-sqr 57.3%

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

      5. swap-sqr 57.3%

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

      6. sqrt-unprod 67.7%

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

      7. add-sqr-sqrt 95.1%

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

      8. *-commutative 95.1%

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

      9. sqr-pow 95.0%

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

      10. metadata-eval 95.0%

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

      11. pow2 95.0%

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

      12. metadata-eval 95.0%

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

      13. pow2 95.0%

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

      14. associate-*r* 94.9%

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

      15. *-commutative 94.9%

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

   10. Applied egg-rr 94.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-90}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 5: 96.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-56}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-90}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.2e-56)
   (* (* x x) (* eps (* 5.0 (* x x))))
   (if (<= x 2.75e-90) (pow eps 5.0) (* (* x x) (* (* eps 5.0) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.2e-56) {
		tmp = (x * x) * (eps * (5.0 * (x * x)));
	} else if (x <= 2.75e-90) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-7.2d-56)) then
        tmp = (x * x) * (eps * (5.0d0 * (x * x)))
    else if (x <= 2.75d-90) then
        tmp = eps ** 5.0d0
    else
        tmp = (x * x) * ((eps * 5.0d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -7.2e-56) {
		tmp = (x * x) * (eps * (5.0 * (x * x)));
	} else if (x <= 2.75e-90) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -7.2e-56:
		tmp = (x * x) * (eps * (5.0 * (x * x)))
	elif x <= 2.75e-90:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (x * x) * ((eps * 5.0) * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7.2e-56)
		tmp = Float64(Float64(x * x) * Float64(eps * Float64(5.0 * Float64(x * x))));
	elseif (x <= 2.75e-90)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * 5.0) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -7.2e-56)
		tmp = (x * x) * (eps * (5.0 * (x * x)));
	elseif (x <= 2.75e-90)
		tmp = eps ^ 5.0;
	else
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.2e-56], N[(N[(x * x), $MachinePrecision] * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.75e-90], N[Power[eps, 5.0], $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.19999999999999956e-56

   1. Initial program 33.3%

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

   2. Taylor expanded in x around inf 97.1%

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

      1. distribute-rgt1-in 97.1%

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

      2. metadata-eval 97.1%

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

   4. Simplified 97.1%

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

      1. add-sqr-sqrt 62.0%

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

      2. sqrt-unprod 41.9%

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

      3. associate-*r* 42.0%

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

      4. *-commutative 42.0%

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

      5. associate-*r* 41.9%

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

      6. *-commutative 41.9%

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

      7. swap-sqr 42.1%

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

      8. *-commutative 42.1%

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

      9. *-commutative 42.1%

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

      10. swap-sqr 42.1%

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

      11. pow-prod-up 42.0%

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

      12. metadata-eval 42.0%

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

      13. metadata-eval 42.0%

          \[\leadsto \sqrt{\left({x}^{8} \cdot \color{blue}{25}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   6. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\sqrt{\left({x}^{8} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}} \]
   7. Step-by-step derivation

      1. metadata-eval 42.0%

         \[\leadsto \sqrt{\left({x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      2. pow-sqr 42.1%

         \[\leadsto \sqrt{\left(\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      3. associate-*l* 42.1%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]

      4. pow-sqr 42.0%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      5. metadata-eval 42.0%

         \[\leadsto \sqrt{{x}^{\color{blue}{8}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   8. Simplified 42.0%

      \[\leadsto \color{blue}{\sqrt{{x}^{8} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]
   9. Step-by-step derivation

      1. metadata-eval 42.0%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      2. pow-sqr 42.1%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      3. metadata-eval 42.1%

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

      4. swap-sqr 42.1%

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

      5. swap-sqr 41.9%

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

      6. sqrt-unprod 62.0%

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

      7. add-sqr-sqrt 97.1%

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

      8. *-commutative 97.1%

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

      9. sqr-pow 97.0%

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

      10. metadata-eval 97.0%

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

      11. pow2 97.0%

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

      12. metadata-eval 97.0%

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

      13. pow2 97.0%

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

      14. associate-*r* 97.0%

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

      15. *-commutative 97.0%

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

   10. Applied egg-rr 97.0%

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

   11. Taylor expanded in eps around 0 96.8%

       \[\leadsto \color{blue}{\left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \cdot \left(x \cdot x\right) \]
   12. Step-by-step derivation

       1. *-commutative 96.8%

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

       2. associate-*l* 97.0%

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

       3. unpow2 97.0%

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

   13. Simplified 97.0%

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

   if -7.19999999999999956e-56 < x < 2.75000000000000015e-90

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.3%

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

   if 2.75000000000000015e-90 < x

   1. Initial program 57.4%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. distribute-rgt1-in 95.1%

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

      2. metadata-eval 95.1%

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

   4. Simplified 95.1%

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

      1. add-sqr-sqrt 67.7%

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

      2. sqrt-unprod 57.3%

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

      3. associate-*r* 57.3%

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

      4. *-commutative 57.3%

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

      5. associate-*r* 57.3%

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

      6. *-commutative 57.3%

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

      7. swap-sqr 57.3%

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

      8. *-commutative 57.3%

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

      9. *-commutative 57.3%

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

      10. swap-sqr 57.3%

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

      11. pow-prod-up 57.3%

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

      12. metadata-eval 57.3%

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

      13. metadata-eval 57.3%

          \[\leadsto \sqrt{\left({x}^{8} \cdot \color{blue}{25}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   6. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\sqrt{\left({x}^{8} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}} \]
   7. Step-by-step derivation

      1. metadata-eval 57.3%

         \[\leadsto \sqrt{\left({x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      2. pow-sqr 57.3%

         \[\leadsto \sqrt{\left(\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      3. associate-*l* 57.3%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]

      4. pow-sqr 57.3%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      5. metadata-eval 57.3%

         \[\leadsto \sqrt{{x}^{\color{blue}{8}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   8. Simplified 57.3%

      \[\leadsto \color{blue}{\sqrt{{x}^{8} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]
   9. Step-by-step derivation

      1. metadata-eval 57.3%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      2. pow-sqr 57.3%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      3. metadata-eval 57.3%

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

      4. swap-sqr 57.3%

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

      5. swap-sqr 57.3%

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

      6. sqrt-unprod 67.7%

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

      7. add-sqr-sqrt 95.1%

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

      8. *-commutative 95.1%

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

      9. sqr-pow 95.0%

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

      10. metadata-eval 95.0%

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

      11. pow2 95.0%

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

      12. metadata-eval 95.0%

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

      13. pow2 95.0%

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

      14. associate-*r* 94.9%

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

      15. *-commutative 94.9%

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

   10. Applied egg-rr 94.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-56}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-90}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 6: 82.2% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in x around inf 85.8%

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

   1. distribute-rgt1-in 85.8%

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

   2. metadata-eval 85.8%

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

4. Simplified 85.8%

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

   1. add-sqr-sqrt 78.6%

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

   2. sqrt-unprod 75.5%

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

   3. associate-*r* 75.5%

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

   4. *-commutative 75.5%

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

   5. associate-*r* 75.5%

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

   6. *-commutative 75.5%

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

   7. swap-sqr 75.5%

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

   8. *-commutative 75.5%

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

   9. *-commutative 75.5%

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

   10. swap-sqr 75.5%

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

   11. pow-prod-up 75.5%

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

   12. metadata-eval 75.5%

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

   13. metadata-eval 75.5%

       \[\leadsto \sqrt{\left({x}^{8} \cdot \color{blue}{25}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

6. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{\sqrt{\left({x}^{8} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}} \]
7. Step-by-step derivation

   1. metadata-eval 75.5%

      \[\leadsto \sqrt{\left({x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   2. pow-sqr 75.5%

      \[\leadsto \sqrt{\left(\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   3. associate-*l* 75.5%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]

   4. pow-sqr 75.5%

      \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   5. metadata-eval 75.5%

      \[\leadsto \sqrt{{x}^{\color{blue}{8}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

8. Simplified 75.5%

   \[\leadsto \color{blue}{\sqrt{{x}^{8} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]
9. Step-by-step derivation

   1. metadata-eval 75.5%

      \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   2. pow-sqr 75.5%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   3. metadata-eval 75.5%

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

   4. swap-sqr 75.5%

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

   5. swap-sqr 75.5%

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

   6. sqrt-unprod 78.6%

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

   7. add-sqr-sqrt 85.8%

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

   8. *-commutative 85.8%

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

   9. sqr-pow 85.8%

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

   10. metadata-eval 85.8%

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

   11. pow2 85.8%

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

   12. metadata-eval 85.8%

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

   13. pow2 85.8%

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

   14. associate-*r* 85.8%

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

   15. *-commutative 85.8%

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

10. Applied egg-rr 85.8%

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

11. Taylor expanded in eps around 0 85.7%

    \[\leadsto \color{blue}{\left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \cdot \left(x \cdot x\right) \]
12. Step-by-step derivation

    1. *-commutative 85.7%

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

    2. associate-*l* 85.7%

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

    3. unpow2 85.7%

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

13. Simplified 85.7%

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

14. Final simplification 85.7%

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

Alternative 7: 82.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in x around inf 85.8%

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

   1. distribute-rgt1-in 85.8%

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

   2. metadata-eval 85.8%

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

4. Simplified 85.8%

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

   1. add-sqr-sqrt 78.6%

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

   2. sqrt-unprod 75.5%

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

   3. associate-*r* 75.5%

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

   4. *-commutative 75.5%

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

   5. associate-*r* 75.5%

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

   6. *-commutative 75.5%

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

   7. swap-sqr 75.5%

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

   8. *-commutative 75.5%

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

   9. *-commutative 75.5%

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

   10. swap-sqr 75.5%

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

   11. pow-prod-up 75.5%

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

   12. metadata-eval 75.5%

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

   13. metadata-eval 75.5%

       \[\leadsto \sqrt{\left({x}^{8} \cdot \color{blue}{25}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

6. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{\sqrt{\left({x}^{8} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}} \]
7. Step-by-step derivation

   1. metadata-eval 75.5%

      \[\leadsto \sqrt{\left({x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   2. pow-sqr 75.5%

      \[\leadsto \sqrt{\left(\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   3. associate-*l* 75.5%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]

   4. pow-sqr 75.5%

      \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   5. metadata-eval 75.5%

      \[\leadsto \sqrt{{x}^{\color{blue}{8}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

8. Simplified 75.5%

   \[\leadsto \color{blue}{\sqrt{{x}^{8} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]
9. Step-by-step derivation

   1. metadata-eval 75.5%

      \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   2. pow-sqr 75.5%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   3. metadata-eval 75.5%

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

   4. swap-sqr 75.5%

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

   5. swap-sqr 75.5%

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

   6. sqrt-unprod 78.6%

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

   7. add-sqr-sqrt 85.8%

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

   8. *-commutative 85.8%

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

   9. sqr-pow 85.8%

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

   10. metadata-eval 85.8%

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

   11. pow2 85.8%

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

   12. metadata-eval 85.8%

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

   13. pow2 85.8%

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

   14. associate-*r* 85.8%

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

   15. *-commutative 85.8%

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

10. Applied egg-rr 85.8%

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

11. Taylor expanded in eps around 0 85.7%

    \[\leadsto \color{blue}{\left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \cdot \left(x \cdot x\right) \]
12. Step-by-step derivation

    1. associate-*r* 85.8%

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

    2. *-commutative 85.8%

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

    3. unpow2 85.8%

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

    4. associate-*l* 85.8%

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

    5. *-commutative 85.8%

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

    6. associate-*l* 85.7%

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

    7. *-commutative 85.7%

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

13. Simplified 85.7%

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

14. Final simplification 85.7%

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

Alternative 8: 82.2% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in x around inf 85.8%

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

   1. distribute-rgt1-in 85.8%

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

   2. metadata-eval 85.8%

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

4. Simplified 85.8%

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

   1. add-sqr-sqrt 78.6%

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

   2. sqrt-unprod 75.5%

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

   3. associate-*r* 75.5%

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

   4. *-commutative 75.5%

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

   5. associate-*r* 75.5%

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

   6. *-commutative 75.5%

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

   7. swap-sqr 75.5%

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

   8. *-commutative 75.5%

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

   9. *-commutative 75.5%

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

   10. swap-sqr 75.5%

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

   11. pow-prod-up 75.5%

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

   12. metadata-eval 75.5%

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

   13. metadata-eval 75.5%

       \[\leadsto \sqrt{\left({x}^{8} \cdot \color{blue}{25}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

6. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{\sqrt{\left({x}^{8} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}} \]
7. Step-by-step derivation

   1. metadata-eval 75.5%

      \[\leadsto \sqrt{\left({x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   2. pow-sqr 75.5%

      \[\leadsto \sqrt{\left(\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot 25\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   3. associate-*l* 75.5%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right) \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]

   4. pow-sqr 75.5%

      \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   5. metadata-eval 75.5%

      \[\leadsto \sqrt{{x}^{\color{blue}{8}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

8. Simplified 75.5%

   \[\leadsto \color{blue}{\sqrt{{x}^{8} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}} \]
9. Step-by-step derivation

   1. metadata-eval 75.5%

      \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot 4\right)}} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   2. pow-sqr 75.5%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{4} \cdot {x}^{4}\right)} \cdot \left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   3. metadata-eval 75.5%

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

   4. swap-sqr 75.5%

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

   5. swap-sqr 75.5%

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

   6. sqrt-unprod 78.6%

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

   7. add-sqr-sqrt 85.8%

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

   8. *-commutative 85.8%

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

   9. sqr-pow 85.8%

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

   10. metadata-eval 85.8%

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

   11. pow2 85.8%

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

   12. metadata-eval 85.8%

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

   13. pow2 85.8%

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

   14. associate-*r* 85.8%

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

   15. *-commutative 85.8%

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

10. Applied egg-rr 85.8%

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

11. Final simplification 85.8%

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

Reproduce

herbie shell --seed 2023275 
(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)))