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

Percentage Accurate: 88.4% → 98.8%

Time: 6.2s

Alternatives: 6

Speedup: 1.9×

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: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, {x}^{2} \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left(10 \cdot {\varepsilon}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-323)
     t_0
     (if (<= t_0 2e-298)
       (fma
        eps
        (* 5.0 (pow x 4.0))
        (*
         (pow x 2.0)
         (+ (* (pow eps 3.0) 10.0) (* x (* 10.0 (pow eps 2.0))))))
       (+ (* x (* 5.0 (pow eps 4.0))) (pow 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 <= -1e-323) {
		tmp = t_0;
	} else if (t_0 <= 2e-298) {
		tmp = fma(eps, (5.0 * pow(x, 4.0)), (pow(x, 2.0) * ((pow(eps, 3.0) * 10.0) + (x * (10.0 * pow(eps, 2.0))))));
	} else {
		tmp = (x * (5.0 * pow(eps, 4.0))) + pow(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 <= -1e-323)
		tmp = t_0;
	elseif (t_0 <= 2e-298)
		tmp = fma(eps, Float64(5.0 * (x ^ 4.0)), Float64((x ^ 2.0) * Float64(Float64((eps ^ 3.0) * 10.0) + Float64(x * Float64(10.0 * (eps ^ 2.0))))));
	else
		tmp = Float64(Float64(x * Float64(5.0 * (eps ^ 4.0))) + (eps ^ 5.0));
	end
	return 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-323], t$95$0, If[LessEqual[t$95$0, 2e-298], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] + N[(x * N[(10.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.88131e-324

   1. Initial program 98.1%

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

   if -9.88131e-324 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 1.99999999999999982e-298

   1. Initial program 85.8%

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

   2. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right) + \left({x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) + {x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, {x}^{2} \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left({\varepsilon}^{2} \cdot 10\right)\right)\right)} \]

   if 1.99999999999999982e-298 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 99.9%

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

   2. Taylor expanded in eps around inf 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-323}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, {x}^{2} \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left(10 \cdot {\varepsilon}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-323)
     t_0
     (if (<= t_0 2e-298)
       (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* 10.0 (pow x 3.0)))))
       (+ (* x (* 5.0 (pow eps 4.0))) (pow 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 <= -1e-323) {
		tmp = t_0;
	} else if (t_0 <= 2e-298) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (10.0 * pow(x, 3.0))));
	} else {
		tmp = (x * (5.0 * pow(eps, 4.0))) + 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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-323)) then
        tmp = t_0
    else if (t_0 <= 2d-298) then
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * (10.0d0 * (x ** 3.0d0))))
    else
        tmp = (x * (5.0d0 * (eps ** 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 <= -1e-323) {
		tmp = t_0;
	} else if (t_0 <= 2e-298) {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (10.0 * Math.pow(x, 3.0))));
	} else {
		tmp = (x * (5.0 * Math.pow(eps, 4.0))) + Math.pow(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 <= -1e-323:
		tmp = t_0
	elif t_0 <= 2e-298:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (10.0 * math.pow(x, 3.0))))
	else:
		tmp = (x * (5.0 * math.pow(eps, 4.0))) + math.pow(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 <= -1e-323)
		tmp = t_0;
	elseif (t_0 <= 2e-298)
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(10.0 * (x ^ 3.0)))));
	else
		tmp = Float64(Float64(x * Float64(5.0 * (eps ^ 4.0))) + (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 <= -1e-323)
		tmp = t_0;
	elseif (t_0 <= 2e-298)
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * (10.0 * (x ^ 3.0))));
	else
		tmp = (x * (5.0 * (eps ^ 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[LessEqual[t$95$0, -1e-323], t$95$0, If[LessEqual[t$95$0, 2e-298], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.88131e-324

   1. Initial program 98.1%

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

   if -9.88131e-324 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 1.99999999999999982e-298

   1. Initial program 85.8%

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

   2. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) + {x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]

   4. Simplified 99.8%

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

   if 1.99999999999999982e-298 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 99.9%

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

   2. Taylor expanded in eps around inf 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-323}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-323} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-323) (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-323) || !(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-323)) .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-323) || !(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-323) 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-323) || !(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-323) || ~((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-323], 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)) < -9.88131e-324 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.3%

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

   if -9.88131e-324 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.8%

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

   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-323} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 4: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-30} \lor \neg \left(x \leq 3.3 \cdot 10^{-53}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-30) (not (<= x 3.3e-53)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-30) || !(x <= 3.3e-53)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.7d-30)) .or. (.not. (x <= 3.3d-53))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-30) || !(x <= 3.3e-53)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.7e-30) or not (x <= 3.3e-53):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.7e-30) || !(x <= 3.3e-53))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.7e-30) || ~((x <= 3.3e-53)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.7e-30], N[Not[LessEqual[x, 3.3e-53]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999987e-30 or 3.30000000000000004e-53 < x

   1. Initial program 27.6%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. distribute-rgt1-in 98.7%

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

      2. metadata-eval 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in x around 0 98.6%

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

   if -2.69999999999999987e-30 < x < 3.30000000000000004e-53

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-30} \lor \neg \left(x \leq 3.3 \cdot 10^{-53}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-30} \lor \neg \left(x \leq 4 \cdot 10^{-54}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-30) (not (<= x 4e-54)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-30) || !(x <= 4e-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 <= (-2.7d-30)) .or. (.not. (x <= 4d-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 <= -2.7e-30) || !(x <= 4e-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 <= -2.7e-30) or not (x <= 4e-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 <= -2.7e-30) || !(x <= 4e-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 <= -2.7e-30) || ~((x <= 4e-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, -2.7e-30], N[Not[LessEqual[x, 4e-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 < -2.69999999999999987e-30 or 4.0000000000000001e-54 < x

   1. Initial program 27.6%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. *-commutative 98.7%

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

      2. distribute-rgt1-in 98.7%

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

      3. metadata-eval 98.7%

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

      4. *-commutative 98.7%

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

      5. associate-*r* 98.8%

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

   4. Simplified 98.8%

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

   if -2.69999999999999987e-30 < x < 4.0000000000000001e-54

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-30} \lor \neg \left(x \leq 4 \cdot 10^{-54}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6: 87.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in x around 0 87.5%

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

3. Final simplification 87.5%

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

Reproduce

herbie shell --seed 2023333 
(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)))