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

Percentage Accurate: 88.5% → 99.3%

Time: 7.2s

Alternatives: 7

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: 88.5% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-306} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -2e-306) (not (<= t_0 0.0)))
     t_0
     (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -2e-306) || !(t_0 <= 0.0)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-2d-306)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    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 <= -2e-306) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	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 <= -2e-306) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -2e-306) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	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 <= -2e-306) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	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, -2e-306], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -2.00000000000000006e-306 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.2%

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

   if -2.00000000000000006e-306 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.2%

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

   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-306} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternative 2: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{-64}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -6e-64)
   (pow eps 5.0)
   (if (<= eps 1.85e-64) (* 5.0 (* eps (pow x 4.0))) (pow eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -6e-64) {
		tmp = pow(eps, 5.0);
	} else if (eps <= 1.85e-64) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-6d-64)) then
        tmp = eps ** 5.0d0
    else if (eps <= 1.85d-64) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -6e-64) {
		tmp = Math.pow(eps, 5.0);
	} else if (eps <= 1.85e-64) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -6e-64:
		tmp = math.pow(eps, 5.0)
	elif eps <= 1.85e-64:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -6e-64)
		tmp = eps ^ 5.0;
	elseif (eps <= 1.85e-64)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -6e-64)
		tmp = eps ^ 5.0;
	elseif (eps <= 1.85e-64)
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -6e-64], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[eps, 1.85e-64], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.0000000000000001e-64 or 1.84999999999999999e-64 < eps

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 93.4%

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

   if -6.0000000000000001e-64 < eps < 1.84999999999999999e-64

   1. Initial program 85.3%

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

   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{-64}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 3: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;\varepsilon \leq 1.06 \cdot 10^{-63}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3e-64)
   (pow eps 5.0)
   (if (<= eps 1.06e-63) (* (pow x 4.0) (* eps 5.0)) (pow eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3e-64) {
		tmp = pow(eps, 5.0);
	} else if (eps <= 1.06e-63) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} 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 (eps <= (-3d-64)) then
        tmp = eps ** 5.0d0
    else if (eps <= 1.06d-63) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3e-64) {
		tmp = Math.pow(eps, 5.0);
	} else if (eps <= 1.06e-63) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3e-64:
		tmp = math.pow(eps, 5.0)
	elif eps <= 1.06e-63:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3e-64)
		tmp = eps ^ 5.0;
	elseif (eps <= 1.06e-63)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3e-64)
		tmp = eps ^ 5.0;
	elseif (eps <= 1.06e-63)
		tmp = (x ^ 4.0) * (eps * 5.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3e-64], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[eps, 1.06e-63], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.0000000000000001e-64 or 1.06000000000000004e-63 < eps

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 93.4%

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

   if -3.0000000000000001e-64 < eps < 1.06000000000000004e-63

   1. Initial program 85.3%

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

   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;\varepsilon \leq 1.06 \cdot 10^{-63}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.7e-64)
   (pow eps 5.0)
   (if (<= eps 2.15e-64) (* (* x x) (* (* eps 5.0) (* x x))) (pow eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.7e-64) {
		tmp = pow(eps, 5.0);
	} else if (eps <= 2.15e-64) {
		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 (eps <= (-1.7d-64)) then
        tmp = eps ** 5.0d0
    else if (eps <= 2.15d-64) 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 (eps <= -1.7e-64) {
		tmp = Math.pow(eps, 5.0);
	} else if (eps <= 2.15e-64) {
		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 eps <= -1.7e-64:
		tmp = math.pow(eps, 5.0)
	elif eps <= 2.15e-64:
		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 (eps <= -1.7e-64)
		tmp = eps ^ 5.0;
	elseif (eps <= 2.15e-64)
		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 (eps <= -1.7e-64)
		tmp = eps ^ 5.0;
	elseif (eps <= 2.15e-64)
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.7e-64], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[eps, 2.15e-64], 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 eps < -1.70000000000000006e-64 or 2.14999999999999987e-64 < eps

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 93.4%

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

   if -1.70000000000000006e-64 < eps < 2.14999999999999987e-64

   1. Initial program 85.3%

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

   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. add-sqr-sqrt 91.6%

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

      2. pow2 91.6%

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

      3. sqrt-prod 54.3%

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

      4. sqrt-pow1 54.3%

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

      5. metadata-eval 54.3%

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

      6. pow2 54.3%

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

      7. *-commutative 54.3%

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

   6. Applied egg-rr 54.3%

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

      1. unpow2 54.3%

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

      2. *-commutative 54.3%

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

      3. *-commutative 54.3%

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

      4. swap-sqr 54.3%

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

      5. add-sqr-sqrt 99.8%

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

      6. associate-*r* 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 83.1% 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 87.8%

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

2. Taylor expanded in x around inf 82.0%

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

   1. distribute-rgt1-in 82.0%

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

   2. metadata-eval 82.0%

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

4. Simplified 82.0%

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

   1. add-sqr-sqrt 75.1%

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

   2. pow2 75.1%

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

   3. sqrt-prod 44.4%

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

   4. sqrt-pow1 44.4%

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

   5. metadata-eval 44.4%

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

   6. pow2 44.4%

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

   7. *-commutative 44.4%

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

6. Applied egg-rr 44.4%

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

   1. unpow2 44.4%

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

   2. *-commutative 44.4%

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

   3. *-commutative 44.4%

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

   4. swap-sqr 44.4%

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

   5. add-sqr-sqrt 82.0%

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

   6. associate-*r* 82.0%

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

8. Applied egg-rr 82.0%

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

9. Taylor expanded in eps around 0 82.0%

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

    1. unpow2 82.0%

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

11. Simplified 82.0%

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

12. Final simplification 82.0%

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

Alternative 6: 83.1% 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 87.8%

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

2. Taylor expanded in x around inf 82.0%

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

   1. distribute-rgt1-in 82.0%

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

   2. metadata-eval 82.0%

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

4. Simplified 82.0%

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

   1. add-sqr-sqrt 75.1%

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

   2. pow2 75.1%

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

   3. sqrt-prod 44.4%

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

   4. sqrt-pow1 44.4%

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

   5. metadata-eval 44.4%

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

   6. pow2 44.4%

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

   7. *-commutative 44.4%

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

6. Applied egg-rr 44.4%

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

   1. unpow2 44.4%

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

   2. *-commutative 44.4%

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

   3. *-commutative 44.4%

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

   4. swap-sqr 44.4%

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

   5. add-sqr-sqrt 82.0%

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

   6. associate-*r* 82.0%

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

8. Applied egg-rr 82.0%

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

9. Final simplification 82.0%

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

Alternative 7: 71.5% 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 87.8%

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

   1. sub-neg 87.8%

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

   2. +-commutative 87.8%

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

   3. sqr-pow 44.9%

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

   4. distribute-rgt-neg-in 44.9%

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

   5. fma-def 44.2%

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

   6. metadata-eval 44.2%

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

   7. metadata-eval 44.2%

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

3. Applied egg-rr 44.2%

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

4. Taylor expanded in eps around 0 69.8%

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

   1. distribute-lft1-in 69.8%

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

   2. metadata-eval 69.8%

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

   3. mul0-lft 69.8%

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

6. Simplified 69.8%

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

7. Final simplification 69.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023293 
(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)))