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

Percentage Accurate: 88.6% → 98.5%

Time: 11.9s

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: 88.6% 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: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-270} \lor \neg \left(t_0 \leq 10^{-308}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) + {\varepsilon}^{3}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-270) (not (<= t_0 1e-308)))
     t_0
     (fma
      (* eps 5.0)
      (pow x 4.0)
      (* x (* x (* 10.0 (+ (* eps (* x eps)) (pow eps 3.0)))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-270) || !(t_0 <= 1e-308)) {
		tmp = t_0;
	} else {
		tmp = fma((eps * 5.0), pow(x, 4.0), (x * (x * (10.0 * ((eps * (x * eps)) + pow(eps, 3.0))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-270) || !(t_0 <= 1e-308))
		tmp = t_0;
	else
		tmp = fma(Float64(eps * 5.0), (x ^ 4.0), Float64(x * Float64(x * Float64(10.0 * Float64(Float64(eps * Float64(x * eps)) + (eps ^ 3.0))))));
	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[Or[LessEqual[t$95$0, -4e-270], N[Not[LessEqual[t$95$0, 1e-308]], $MachinePrecision]], t$95$0, N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(x * N[(x * N[(10.0 * N[(N[(eps * N[(x * eps), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.0000000000000002e-270 or 9.9999999999999991e-309 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 96.3%

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

   if -4.0000000000000002e-270 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 9.9999999999999991e-309

   1. Initial program 87.9%

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

   2. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. fma-def 99.8%

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

      2. distribute-lft1-in 99.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 99.8%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 99.8%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 99.8%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 99.8%

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

      2. associate-*r* 99.8%

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

      3. unpow2 99.8%

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

      4. cube-mult 99.8%

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

      5. associate-*r* 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. distribute-rgt-in 99.8%

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

      9. fma-udef 99.8%

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

      10. associate-*l* 99.8%

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

      11. fma-udef 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-*r* 99.8%

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

      14. unpow2 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-270} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 10^{-308}\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) + {\varepsilon}^{3}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-270} \lor \neg \left(t_0 \leq 10^{-308}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-270) (not (<= t_0 1e-308)))
     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 <= -4e-270) || !(t_0 <= 1e-308)) {
		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 <= (-4d-270)) .or. (.not. (t_0 <= 1d-308))) 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 <= -4e-270) || !(t_0 <= 1e-308)) {
		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 <= -4e-270) or not (t_0 <= 1e-308):
		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 <= -4e-270) || !(t_0 <= 1e-308))
		tmp = t_0;
	else
		tmp = Float64(Float64(eps * 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 <= -4e-270) || ~((t_0 <= 1e-308)))
		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, -4e-270], N[Not[LessEqual[t$95$0, 1e-308]], $MachinePrecision]], t$95$0, N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.0000000000000002e-270 or 9.9999999999999991e-309 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 96.3%

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

   if -4.0000000000000002e-270 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 9.9999999999999991e-309

   1. Initial program 87.9%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. distribute-lft1-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-270} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 10^{-308}\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 3: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{-59} \lor \neg \left(x \leq 1.22 \cdot 10^{-48}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.86e-59) (not (<= x 1.22e-48)))
   (* eps (+ (* (* x x) (* 5.0 (* x x))) (* eps (* 10.0 (pow x 3.0)))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.86e-59) || !(x <= 1.22e-48)) {
		tmp = eps * (((x * x) * (5.0 * (x * x))) + (eps * (10.0 * pow(x, 3.0))));
	} 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 <= (-1.86d-59)) .or. (.not. (x <= 1.22d-48))) then
        tmp = eps * (((x * x) * (5.0d0 * (x * x))) + (eps * (10.0d0 * (x ** 3.0d0))))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.86e-59) || !(x <= 1.22e-48)) {
		tmp = eps * (((x * x) * (5.0 * (x * x))) + (eps * (10.0 * Math.pow(x, 3.0))));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.86e-59) or not (x <= 1.22e-48):
		tmp = eps * (((x * x) * (5.0 * (x * x))) + (eps * (10.0 * math.pow(x, 3.0))))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.86e-59) || !(x <= 1.22e-48))
		tmp = Float64(eps * Float64(Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))) + Float64(eps * Float64(10.0 * (x ^ 3.0)))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.86e-59) || ~((x <= 1.22e-48)))
		tmp = eps * (((x * x) * (5.0 * (x * x))) + (eps * (10.0 * (x ^ 3.0))));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.86e-59], N[Not[LessEqual[x, 1.22e-48]], $MachinePrecision]], N[(eps * N[(N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.86000000000000004e-59 or 1.21999999999999993e-48 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in eps around 0 90.9%

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x\right) + \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 90.9%

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

      2. unpow2 90.9%

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

      3. associate-*l* 90.9%

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

      4. distribute-lft-out 90.9%

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

      5. distribute-lft1-in 90.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(4 + 1\right) \cdot {x}^{4}} + \varepsilon \cdot \left(4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x\right)\right) \]

      6. metadata-eval 90.9%

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

      7. *-commutative 90.9%

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

      8. *-commutative 90.9%

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

      9. distribute-rgt-out 90.9%

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

      10. associate-*r* 90.9%

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

   4. Simplified 90.9%

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

      1. add-sqr-sqrt 90.8%

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

      2. pow2 90.8%

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

      3. *-commutative 90.8%

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

      4. sqrt-prod 90.6%

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

      5. sqrt-pow1 90.6%

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

      6. metadata-eval 90.6%

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

      7. pow2 90.6%

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

   6. Applied egg-rr 90.6%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{{\left(\left(x \cdot x\right) \cdot \sqrt{5}\right)}^{2}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.6%

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

      2. swap-sqr 90.4%

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

      3. add-sqr-sqrt 90.9%

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

      4. associate-*l* 90.8%

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

   8. Applied egg-rr 90.8%

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

   if -1.86000000000000004e-59 < x < 1.21999999999999993e-48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{-59} \lor \neg \left(x \leq 1.22 \cdot 10^{-48}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-60} \lor \neg \left(x \leq 1.7 \cdot 10^{-48}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7.5e-60) (not (<= x 1.7e-48)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7.5e-60) || !(x <= 1.7e-48)) {
		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 <= (-7.5d-60)) .or. (.not. (x <= 1.7d-48))) 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 <= -7.5e-60) || !(x <= 1.7e-48)) {
		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 <= -7.5e-60) or not (x <= 1.7e-48):
		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 <= -7.5e-60) || !(x <= 1.7e-48))
		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 <= -7.5e-60) || ~((x <= 1.7e-48)))
		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, -7.5e-60], N[Not[LessEqual[x, 1.7e-48]], $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 < -7.5000000000000002e-60 or 1.70000000000000014e-48 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in x around inf 89.1%

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

      1. distribute-lft1-in 89.1%

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

      2. metadata-eval 89.1%

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

      3. associate-*l* 89.0%

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

   4. Simplified 89.0%

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

   if -7.5000000000000002e-60 < x < 1.70000000000000014e-48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-60} \lor \neg \left(x \leq 1.7 \cdot 10^{-48}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-59} \lor \neg \left(x \leq 3.9 \cdot 10^{-48}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.7e-59) (not (<= x 3.9e-48)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.7e-59) || !(x <= 3.9e-48)) {
		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-59)) .or. (.not. (x <= 3.9d-48))) 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-59) || !(x <= 3.9e-48)) {
		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-59) or not (x <= 3.9e-48):
		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-59) || !(x <= 3.9e-48))
		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-59) || ~((x <= 3.9e-48)))
		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-59], N[Not[LessEqual[x, 3.9e-48]], $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 < -1.70000000000000009e-59 or 3.9e-48 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in eps around 0 89.1%

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

   3. Taylor expanded in x around 0 89.1%

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

   if -1.70000000000000009e-59 < x < 3.9e-48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-59} \lor \neg \left(x \leq 3.9 \cdot 10^{-48}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{-59} \lor \neg \left(x \leq 5.2 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.86e-59) (not (<= x 5.2e-48)))
   (* (* eps 5.0) (pow x 4.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.86e-59) || !(x <= 5.2e-48)) {
		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 <= (-1.86d-59)) .or. (.not. (x <= 5.2d-48))) 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 <= -1.86e-59) || !(x <= 5.2e-48)) {
		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 <= -1.86e-59) or not (x <= 5.2e-48):
		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 <= -1.86e-59) || !(x <= 5.2e-48))
		tmp = Float64(Float64(eps * 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 <= -1.86e-59) || ~((x <= 5.2e-48)))
		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, -1.86e-59], N[Not[LessEqual[x, 5.2e-48]], $MachinePrecision]], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.86000000000000004e-59 or 5.19999999999999975e-48 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in x around inf 89.1%

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

      1. distribute-lft1-in 89.1%

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

      2. metadata-eval 89.1%

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

      3. *-commutative 89.1%

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

   4. Simplified 89.1%

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

   if -1.86000000000000004e-59 < x < 5.19999999999999975e-48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{-59} \lor \neg \left(x \leq 5.2 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 7: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{-59} \lor \neg \left(x \leq 1.15 \cdot 10^{-48}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.86e-59) (not (<= x 1.15e-48)))
   (* (* x x) (* (* eps 5.0) (* x x)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.86e-59) || !(x <= 1.15e-48)) {
		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 <= (-1.86d-59)) .or. (.not. (x <= 1.15d-48))) 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 <= -1.86e-59) || !(x <= 1.15e-48)) {
		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 <= -1.86e-59) or not (x <= 1.15e-48):
		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 <= -1.86e-59) || !(x <= 1.15e-48))
		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 <= -1.86e-59) || ~((x <= 1.15e-48)))
		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, -1.86e-59], N[Not[LessEqual[x, 1.15e-48]], $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 < -1.86000000000000004e-59 or 1.15e-48 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in x around inf 89.1%

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

      1. distribute-lft1-in 89.1%

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

      2. metadata-eval 89.1%

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

      3. associate-*l* 89.0%

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

   4. Simplified 89.0%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

      3. add-sqr-sqrt 66.6%

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

      4. pow2 66.7%

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

      5. *-commutative 66.7%

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

      6. sqrt-prod 55.8%

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

      7. sqrt-pow1 55.8%

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

      8. metadata-eval 55.8%

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

      9. pow2 55.8%

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

      10. *-commutative 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. unpow2 55.8%

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

      2. *-commutative 55.8%

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

      3. *-commutative 55.8%

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

      4. swap-sqr 55.9%

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

      5. add-sqr-sqrt 88.9%

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

      6. associate-*r* 88.8%

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

      7. *-commutative 88.8%

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

   8. Applied egg-rr 88.8%

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

   if -1.86000000000000004e-59 < x < 1.15e-48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{-59} \lor \neg \left(x \leq 1.15 \cdot 10^{-48}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 8: 82.2% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.5%

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

2. Taylor expanded in x around inf 82.5%

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

   1. distribute-lft1-in 82.5%

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

   2. metadata-eval 82.5%

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

   3. associate-*l* 82.5%

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

4. Simplified 82.5%

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

   1. associate-*r* 82.5%

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

   2. *-commutative 82.5%

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

   3. add-sqr-sqrt 77.6%

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

   4. pow2 77.6%

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

   5. *-commutative 77.6%

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

   6. sqrt-prod 40.8%

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

   7. sqrt-pow1 40.8%

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

   8. metadata-eval 40.8%

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

   9. pow2 40.8%

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

   10. *-commutative 40.8%

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

6. Applied egg-rr 40.8%

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

   1. unpow2 40.8%

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

   2. *-commutative 40.8%

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

   3. *-commutative 40.8%

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

   4. swap-sqr 40.9%

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

   5. add-sqr-sqrt 82.5%

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

   6. associate-*r* 82.5%

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

   7. *-commutative 82.5%

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

8. Applied egg-rr 82.5%

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

9. Final simplification 82.5%

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

Reproduce

herbie shell --seed 2023187 
(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)))