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

Percentage Accurate: 88.9% → 97.2%

Time: 5.9s

Alternatives: 5

Speedup: 1.8×

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: 88.9% 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: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-91}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot {x}^{3}, \varepsilon \cdot 10, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (* eps (pow x 4.0)))))
   (if (<= x -5.5e-47)
     t_0
     (if (<= x 2.35e-91)
       (- (pow (+ x eps) 5.0) (pow x 5.0))
       (fma (* eps (pow x 3.0)) (* eps 10.0) t_0)))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * (eps * pow(x, 4.0));
	double tmp;
	if (x <= -5.5e-47) {
		tmp = t_0;
	} else if (x <= 2.35e-91) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = fma((eps * pow(x, 3.0)), (eps * 10.0), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(5.0 * Float64(eps * (x ^ 4.0)))
	tmp = 0.0
	if (x <= -5.5e-47)
		tmp = t_0;
	elseif (x <= 2.35e-91)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = fma(Float64(eps * (x ^ 3.0)), Float64(eps * 10.0), t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e-47], t$95$0, If[LessEqual[x, 2.35e-91], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(eps * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * 10.0), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5000000000000002e-47

   1. Initial program 10.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around 0 99.8%

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

   if -5.5000000000000002e-47 < x < 2.35000000000000003e-91

   1. Initial program 99.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   if 2.35000000000000003e-91 < x

   1. Initial program 57.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.7%

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

   5. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. associate-*r* 99.7%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. associate-*r* 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*l* 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot {x}^{3}, 10 \cdot \varepsilon, 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-91}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot {x}^{3}, \varepsilon \cdot 10, 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-91}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.05e-47)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 2.35e-91)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* (pow x 3.0) 10.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.05e-47) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 2.35e-91) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (pow(x, 3.0) * 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.05d-47)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 2.35d-91) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * ((x ** 3.0d0) * 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.05e-47) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 2.35e-91) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (Math.pow(x, 3.0) * 10.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.05e-47:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 2.35e-91:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (math.pow(x, 3.0) * 10.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.05e-47)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 2.35e-91)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64((x ^ 3.0) * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.05e-47)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 2.35e-91)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * ((x ^ 3.0) * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.05e-47], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.35e-91], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.05000000000000001e-47

   1. Initial program 10.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around 0 99.8%

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

   if -2.05000000000000001e-47 < x < 2.35000000000000003e-91

   1. Initial program 99.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   if 2.35000000000000003e-91 < x

   1. Initial program 57.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.7%

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-91}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-46} \lor \neg \left(x \leq 2.35 \cdot 10^{-91}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -9.5e-46) (not (<= x 2.35e-91)))
   (* 5.0 (* eps (pow x 4.0)))
   (- (pow (+ x eps) 5.0) (pow x 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -9.5e-46) || !(x <= 2.35e-91)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow((x + eps), 5.0) - pow(x, 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 <= (-9.5d-46)) .or. (.not. (x <= 2.35d-91))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -9.5e-46) || !(x <= 2.35e-91)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -9.5e-46) or not (x <= 2.35e-91):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -9.5e-46) || !(x <= 2.35e-91))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -9.5e-46) || ~((x <= 2.35e-91)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -9.5e-46], N[Not[LessEqual[x, 2.35e-91]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999993e-46 or 2.35000000000000003e-91 < x

   1. Initial program 42.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.6%

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

   if -9.49999999999999993e-46 < x < 2.35000000000000003e-91

   1. Initial program 99.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-46} \lor \neg \left(x \leq 2.35 \cdot 10^{-91}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-48} \lor \neg \left(x \leq 2.35 \cdot 10^{-91}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.5e-48) (not (<= x 2.35e-91)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.5e-48) || !(x <= 2.35e-91)) {
		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 <= (-5.5d-48)) .or. (.not. (x <= 2.35d-91))) 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 <= -5.5e-48) || !(x <= 2.35e-91)) {
		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 <= -5.5e-48) or not (x <= 2.35e-91):
		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 <= -5.5e-48) || !(x <= 2.35e-91))
		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 <= -5.5e-48) || ~((x <= 2.35e-91)))
		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, -5.5e-48], N[Not[LessEqual[x, 2.35e-91]], $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 < -5.50000000000000047e-48 or 2.35000000000000003e-91 < x

   1. Initial program 42.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.6%

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

   if -5.50000000000000047e-48 < x < 2.35000000000000003e-91

   1. Initial program 99.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-48} \lor \neg \left(x \leq 2.35 \cdot 10^{-91}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% 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 85.4%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 85.3%

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

4. Final simplification 85.3%

   \[\leadsto {\varepsilon}^{5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))