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

Percentage Accurate: 88.4% → 99.1%

Time: 7.7s

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: 88.4% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-318)
     (+ (pow eps 5.0) (* x (* 5.0 (pow eps 4.0))))
     (if (<= t_0 0.0) (* eps (* 5.0 (pow x 4.0))) t_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 <= -1e-318) {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = 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 + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-318)) then
        tmp = (eps ** 5.0d0) + (x * (5.0d0 * (eps ** 4.0d0)))
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_0
    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 <= -1e-318) {
		tmp = Math.pow(eps, 5.0) + (x * (5.0 * Math.pow(eps, 4.0)));
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_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 <= -1e-318:
		tmp = math.pow(eps, 5.0) + (x * (5.0 * math.pow(eps, 4.0)))
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_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 <= -1e-318)
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_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 <= -1e-318)
		tmp = (eps ^ 5.0) + (x * (5.0 * (eps ^ 4.0)));
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_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[LessEqual[t$95$0, -1e-318], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.9999875e-319

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 87.0%

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

   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%

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

   5. Taylor expanded in eps around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

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

   1. Initial program 96.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-318} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-318) (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 <= -1e-318) || !(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 <= (-1d-318)) .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 <= -1e-318) || !(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 <= -1e-318) 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 <= -1e-318) || !(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 <= -1e-318) || ~((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, -1e-318], 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)) < -9.9999875e-319 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.4%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 87.0%

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

   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%

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

   5. Taylor expanded in eps around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 3: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-49} \lor \neg \left(x \leq 5.8 \cdot 10^{-54}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.7e-49) (not (<= x 5.8e-54)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.7e-49) || !(x <= 5.8e-54)) {
		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 <= (-1.7d-49)) .or. (.not. (x <= 5.8d-54))) 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 <= -1.7e-49) || !(x <= 5.8e-54)) {
		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 <= -1.7e-49) or not (x <= 5.8e-54):
		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 <= -1.7e-49) || !(x <= 5.8e-54))
		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 <= -1.7e-49) || ~((x <= 5.8e-54)))
		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, -1.7e-49], N[Not[LessEqual[x, 5.8e-54]], $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 < -1.70000000000000002e-49 or 5.80000000000000029e-54 < x

   1. Initial program 30.4%

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

   2. Taylor expanded in x around inf 92.7%

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

      1. distribute-lft1-in 92.7%

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

      2. metadata-eval 92.7%

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

      3. *-commutative 92.7%

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

   4. Simplified 92.7%

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

   5. Taylor expanded in eps around 0 92.6%

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

      1. associate-*r* 92.7%

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

      2. *-commutative 92.7%

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

      3. associate-*r* 92.9%

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

   7. Simplified 92.9%

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

   if -1.70000000000000002e-49 < x < 5.80000000000000029e-54

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-49} \lor \neg \left(x \leq 5.8 \cdot 10^{-54}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-49}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.2e-49)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 3.7e-53) (pow eps 5.0) (* (* x x) (* (* eps 5.0) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-49) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 3.7e-53) {
		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 <= (-2.2d-49)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 3.7d-53) 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 <= -2.2e-49) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 3.7e-53) {
		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 <= -2.2e-49:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 3.7e-53:
		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 <= -2.2e-49)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 3.7e-53)
		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 <= -2.2e-49)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 3.7e-53)
		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, -2.2e-49], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e-53], 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 < -2.1999999999999999e-49

   1. Initial program 27.4%

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

   2. Taylor expanded in x around inf 89.8%

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

      1. distribute-lft1-in 89.8%

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

      2. metadata-eval 89.8%

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

      3. associate-*l* 89.7%

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

   4. Simplified 89.7%

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

   if -2.1999999999999999e-49 < x < 3.69999999999999982e-53

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 3.69999999999999982e-53 < x

   1. Initial program 33.5%

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

   2. Taylor expanded in x around inf 95.6%

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

      1. distribute-lft1-in 95.6%

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

      2. metadata-eval 95.6%

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

      3. *-commutative 95.6%

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

   4. Simplified 95.6%

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

      1. add-sqr-sqrt 70.8%

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

      2. sqrt-unprod 53.6%

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

      3. *-commutative 53.6%

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

      4. *-commutative 53.6%

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

      5. swap-sqr 53.6%

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

      6. pow-prod-up 53.6%

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

      7. metadata-eval 53.6%

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

      8. swap-sqr 53.6%

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

      9. metadata-eval 53.6%

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

   6. Applied egg-rr 53.6%

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

      1. *-commutative 53.6%

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

      2. sqrt-prod 62.6%

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

      3. sqrt-pow1 62.6%

         \[\leadsto \sqrt{\left(\varepsilon \cdot \varepsilon\right) \cdot 25} \cdot \color{blue}{{x}^{\left(\frac{8}{2}\right)}} \]

      4. metadata-eval 62.6%

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

      5. metadata-eval 62.6%

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

      6. pow-sqr 62.6%

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

      7. pow-prod-down 62.6%

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

      8. pow2 62.6%

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

      9. associate-*r* 62.5%

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

      10. *-commutative 62.5%

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

      11. sqrt-prod 62.5%

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

      12. metadata-eval 62.5%

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

      13. sqrt-prod 60.5%

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

      14. add-sqr-sqrt 95.6%

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

   8. Applied egg-rr 95.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-49}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 5: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-49} \lor \neg \left(x \leq 3.7 \cdot 10^{-54}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.2e-49) (not (<= x 3.7e-54)))
   (* (* x x) (* (* eps 5.0) (* x x)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.2e-49) || !(x <= 3.7e-54)) {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	} 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 <= (-2.2d-49)) .or. (.not. (x <= 3.7d-54))) then
        tmp = (x * x) * ((eps * 5.0d0) * (x * x))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.2e-49) || !(x <= 3.7e-54)) {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.2e-49) or not (x <= 3.7e-54):
		tmp = (x * x) * ((eps * 5.0) * (x * x))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.2e-49) || !(x <= 3.7e-54))
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * 5.0) * Float64(x * x)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.2e-49) || ~((x <= 3.7e-54)))
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.2e-49], N[Not[LessEqual[x, 3.7e-54]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e-49 or 3.7000000000000003e-54 < x

   1. Initial program 30.4%

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

   2. Taylor expanded in x around inf 92.7%

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

      1. distribute-lft1-in 92.7%

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

      2. metadata-eval 92.7%

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

      3. *-commutative 92.7%

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

   4. Simplified 92.7%

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

      1. add-sqr-sqrt 62.7%

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

      2. sqrt-unprod 41.8%

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

      3. *-commutative 41.8%

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

      4. *-commutative 41.8%

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

      5. swap-sqr 39.5%

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

      6. pow-prod-up 39.5%

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

      7. metadata-eval 39.5%

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

      8. swap-sqr 39.5%

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

      9. metadata-eval 39.5%

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

   6. Applied egg-rr 39.5%

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

      1. *-commutative 39.5%

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

      2. sqrt-prod 48.7%

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

      3. sqrt-pow1 55.5%

         \[\leadsto \sqrt{\left(\varepsilon \cdot \varepsilon\right) \cdot 25} \cdot \color{blue}{{x}^{\left(\frac{8}{2}\right)}} \]

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

      6. pow-sqr 55.4%

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

      7. pow-prod-down 55.4%

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

      8. pow2 55.4%

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

      9. associate-*r* 55.4%

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

      10. *-commutative 55.4%

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

      11. sqrt-prod 55.4%

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

      12. metadata-eval 55.4%

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

      13. sqrt-prod 57.5%

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

      14. add-sqr-sqrt 92.4%

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

   8. Applied egg-rr 92.4%

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

   if -2.1999999999999999e-49 < x < 3.7000000000000003e-54

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-49} \lor \neg \left(x \leq 3.7 \cdot 10^{-54}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6: 82.8% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in x around inf 83.5%

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

   1. distribute-lft1-in 83.5%

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

   2. metadata-eval 83.5%

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

   3. *-commutative 83.5%

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

4. Simplified 83.5%

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

   1. add-sqr-sqrt 78.8%

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

   2. sqrt-unprod 75.5%

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

   3. *-commutative 75.5%

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

   4. *-commutative 75.5%

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

   5. swap-sqr 75.2%

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

   6. pow-prod-up 75.2%

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

   7. metadata-eval 75.2%

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

   8. swap-sqr 75.2%

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

   9. metadata-eval 75.2%

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

6. Applied egg-rr 75.2%

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

   1. *-commutative 75.2%

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

   2. sqrt-prod 76.6%

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

   3. sqrt-pow1 77.7%

      \[\leadsto \sqrt{\left(\varepsilon \cdot \varepsilon\right) \cdot 25} \cdot \color{blue}{{x}^{\left(\frac{8}{2}\right)}} \]

   4. metadata-eval 77.7%

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

   5. metadata-eval 77.7%

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

   6. pow-sqr 77.7%

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

   7. pow-prod-down 77.7%

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

   8. pow2 77.7%

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

   9. associate-*r* 77.7%

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

   10. *-commutative 77.7%

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

   11. sqrt-prod 77.7%

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

   12. metadata-eval 77.7%

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

   13. sqrt-prod 39.5%

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

   14. add-sqr-sqrt 83.5%

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

8. Applied egg-rr 83.5%

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

9. Taylor expanded in eps around 0 83.4%

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

    1. unpow2 83.4%

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

11. Simplified 83.4%

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

12. Final simplification 83.4%

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

Alternative 7: 82.8% 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 89.1%

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

2. Taylor expanded in x around inf 83.5%

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

   1. distribute-lft1-in 83.5%

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

   2. metadata-eval 83.5%

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

   3. *-commutative 83.5%

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

4. Simplified 83.5%

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

   1. add-sqr-sqrt 78.8%

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

   2. sqrt-unprod 75.5%

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

   3. *-commutative 75.5%

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

   4. *-commutative 75.5%

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

   5. swap-sqr 75.2%

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

   6. pow-prod-up 75.2%

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

   7. metadata-eval 75.2%

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

   8. swap-sqr 75.2%

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

   9. metadata-eval 75.2%

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

6. Applied egg-rr 75.2%

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

   1. *-commutative 75.2%

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

   2. sqrt-prod 76.6%

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

   3. sqrt-pow1 77.7%

      \[\leadsto \sqrt{\left(\varepsilon \cdot \varepsilon\right) \cdot 25} \cdot \color{blue}{{x}^{\left(\frac{8}{2}\right)}} \]

   4. metadata-eval 77.7%

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

   5. metadata-eval 77.7%

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

   6. pow-sqr 77.7%

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

   7. pow-prod-down 77.7%

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

   8. pow2 77.7%

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

   9. associate-*r* 77.7%

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

   10. *-commutative 77.7%

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

   11. sqrt-prod 77.7%

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

   12. metadata-eval 77.7%

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

   13. sqrt-prod 39.5%

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

   14. add-sqr-sqrt 83.5%

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

8. Applied egg-rr 83.5%

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

9. Final simplification 83.5%

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

Alternative 8: 71.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 89.1%

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

   1. sqr-pow 41.7%

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

   2. metadata-eval 41.7%

      \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)} - {x}^{5} \]

   3. metadata-eval 41.7%

      \[\leadsto {\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} - {x}^{5} \]

3. Applied egg-rr 41.7%

   \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{2.5}} - {x}^{5} \]
4. Step-by-step derivation

   1. pow-prod-down 76.4%

      \[\leadsto \color{blue}{{\left(\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)}^{2.5}} - {x}^{5} \]

   2. pow2 76.4%

      \[\leadsto {\color{blue}{\left({\left(x + \varepsilon\right)}^{2}\right)}}^{2.5} - {x}^{5} \]

   3. +-commutative 76.4%

      \[\leadsto {\left({\color{blue}{\left(\varepsilon + x\right)}}^{2}\right)}^{2.5} - {x}^{5} \]

5. Applied egg-rr 76.4%

   \[\leadsto \color{blue}{{\left({\left(\varepsilon + x\right)}^{2}\right)}^{2.5}} - {x}^{5} \]

6. Taylor expanded in x around inf 72.6%

   \[\leadsto \color{blue}{{x}^{5} + -1 \cdot {x}^{5}} \]
7. Step-by-step derivation

   1. distribute-rgt1-in 72.6%

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

   2. metadata-eval 72.6%

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

   3. mul0-lft 72.6%

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

8. Simplified 72.6%

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

9. Final simplification 72.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023257 
(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)))