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

Percentage Accurate: 88.9% → 98.9%

Time: 8.7s

Alternatives: 12

Speedup: 5.4×

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: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-318)
     t_0
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-318) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((eps + x) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-4d-318)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((eps + x), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-318) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((eps + x), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -4e-318:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-318)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((eps + x) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -4e-318)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-318], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.9999999e-318

   1. Initial program 95.1%

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

   if -3.9999999e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.5%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(10, x \cdot x, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot {\varepsilon}^{3}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-318)
     (* (fma 10.0 (* x x) (* (fma 5.0 x eps) eps)) (pow eps 3.0))
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-318) {
		tmp = fma(10.0, (x * x), (fma(5.0, x, eps) * eps)) * pow(eps, 3.0);
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-318)
		tmp = Float64(fma(10.0, Float64(x * x), Float64(fma(5.0, x, eps) * eps)) * (eps ^ 3.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-318], N[(N[(10.0 * N[(x * x), $MachinePrecision] + N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.9999999e-318

   1. Initial program 95.1%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right) \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \color{blue}{-1}\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 90.6%

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

   if -3.9999999e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(10, x \cdot x, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot {\varepsilon}^{3}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-318)
     (* (* (fma (* x x) 10.0 (* (fma 5.0 x eps) eps)) (* eps eps)) eps)
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-318) {
		tmp = (fma((x * x), 10.0, (fma(5.0, x, eps) * eps)) * (eps * eps)) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-318)
		tmp = Float64(Float64(fma(Float64(x * x), 10.0, Float64(fma(5.0, x, eps) * eps)) * Float64(eps * eps)) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-318], N[(N[(N[(N[(x * x), $MachinePrecision] * 10.0 + N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.9999999e-318

   1. Initial program 95.1%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right) \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \color{blue}{-1}\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.9%

      \[\leadsto \left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 90.6%

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

   13. Applied rewrites 90.4%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   if -3.9999999e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.1%

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

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-318)
     (* (* (fma (* x x) 10.0 (* (fma 5.0 x eps) eps)) (* eps eps)) eps)
     (if (<= t_0 0.0) (* (* (* x x) eps) (* (* x x) 5.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-318) {
		tmp = (fma((x * x), 10.0, (fma(5.0, x, eps) * eps)) * (eps * eps)) * eps;
	} else if (t_0 <= 0.0) {
		tmp = ((x * x) * eps) * ((x * x) * 5.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-318)
		tmp = Float64(Float64(fma(Float64(x * x), 10.0, Float64(fma(5.0, x, eps) * eps)) * Float64(eps * eps)) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(x * x) * eps) * Float64(Float64(x * x) * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-318], N[(N[(N[(N[(x * x), $MachinePrecision] * 10.0 + N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.9999999e-318

   1. Initial program 95.1%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right) \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \color{blue}{-1}\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.9%

      \[\leadsto \left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 90.6%

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

   13. Applied rewrites 90.4%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   if -3.9999999e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 88.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.0%

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

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-318)
     (* (* (fma (* x x) 10.0 (* (fma 5.0 x eps) eps)) (* eps eps)) eps)
     (if (<= t_0 0.0)
       (* (* (* x x) eps) (* (* x x) 5.0))
       (* (* (* eps eps) (* eps eps)) (fma 5.0 x eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-318) {
		tmp = (fma((x * x), 10.0, (fma(5.0, x, eps) * eps)) * (eps * eps)) * eps;
	} else if (t_0 <= 0.0) {
		tmp = ((x * x) * eps) * ((x * x) * 5.0);
	} else {
		tmp = ((eps * eps) * (eps * eps)) * fma(5.0, x, eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-318)
		tmp = Float64(Float64(fma(Float64(x * x), 10.0, Float64(fma(5.0, x, eps) * eps)) * Float64(eps * eps)) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(x * x) * eps) * Float64(Float64(x * x) * 5.0));
	else
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * fma(5.0, x, eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-318], N[(N[(N[(N[(x * x), $MachinePrecision] * 10.0 + N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.9999999e-318

   1. Initial program 95.1%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right) \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \color{blue}{-1}\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right)} \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.9%

      \[\leadsto \left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 90.6%

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

   13. Applied rewrites 90.4%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   if -3.9999999e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 88.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.8%

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

   10. Applied rewrites 99.4%

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

4. Final simplification 99.0%

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

Alternative 6: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon, \frac{-10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{-x}\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.5e-50)
   (* (* (* x x) (* x x)) (* 5.0 eps))
   (if (<= x 1.95e-35)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (* (fma 5.0 eps (/ (* -10.0 (* eps eps)) (- x))) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-50) {
		tmp = ((x * x) * (x * x)) * (5.0 * eps);
	} else if (x <= 1.95e-35) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else {
		tmp = fma(5.0, eps, ((-10.0 * (eps * eps)) / -x)) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.5e-50)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(5.0 * eps));
	elseif (x <= 1.95e-35)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(fma(5.0, eps, Float64(Float64(-10.0 * Float64(eps * eps)) / Float64(-x))) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.5e-50], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e-35], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * eps + N[(N[(-10.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.49999999999999995e-50

   1. Initial program 32.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 32.2

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 32.2

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

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

   if -1.49999999999999995e-50 < x < 1.9499999999999999e-35

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

   if 1.9499999999999999e-35 < x

   1. Initial program 21.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon, \frac{-10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{-x}\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-35}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.5e-50)
   (* (* (* x x) (* x x)) (* 5.0 eps))
   (if (<= x 1.95e-35)
     (* (* (* eps eps) (* eps eps)) (fma 5.0 x eps))
     (* (* (* x x) eps) (* (* x x) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-50) {
		tmp = ((x * x) * (x * x)) * (5.0 * eps);
	} else if (x <= 1.95e-35) {
		tmp = ((eps * eps) * (eps * eps)) * fma(5.0, x, eps);
	} else {
		tmp = ((x * x) * eps) * ((x * x) * 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.5e-50)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(5.0 * eps));
	elseif (x <= 1.95e-35)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * fma(5.0, x, eps));
	else
		tmp = Float64(Float64(Float64(x * x) * eps) * Float64(Float64(x * x) * 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.5e-50], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e-35], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.49999999999999995e-50

   1. Initial program 32.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 32.2

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 32.2

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

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

   if -1.49999999999999995e-50 < x < 1.9499999999999999e-35

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.4%

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

   if 1.9499999999999999e-35 < x

   1. Initial program 21.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 21.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 21.6

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 91.7

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 91.4%

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

   11. Applied rewrites 91.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-35}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-35}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.5e-50)
   (* (* (* x x) (* x x)) (* 5.0 eps))
   (if (<= x 1.95e-35)
     (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
     (* (* (* x x) eps) (* (* x x) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-50) {
		tmp = ((x * x) * (x * x)) * (5.0 * eps);
	} else if (x <= 1.95e-35) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = ((x * x) * eps) * ((x * x) * 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.5e-50)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(5.0 * eps));
	elseif (x <= 1.95e-35)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = Float64(Float64(Float64(x * x) * eps) * Float64(Float64(x * x) * 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.5e-50], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e-35], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.49999999999999995e-50

   1. Initial program 32.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 32.2

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 32.2

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

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

   if -1.49999999999999995e-50 < x < 1.9499999999999999e-35

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 99.5

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.3%

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

   if 1.9499999999999999e-35 < x

   1. Initial program 21.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 21.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 21.6

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 91.7

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 91.4%

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

   11. Applied rewrites 91.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-35}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.1% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 89.4

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 89.4

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

4. Applied rewrites 89.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 85.1

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

7. Applied rewrites 85.1%

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

9. Applied rewrites 85.1%

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

11. Applied rewrites 85.1%

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

12. Final simplification 85.1%

    \[\leadsto \left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right) \]
13. Add Preprocessing

Alternative 10: 83.1% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 89.4

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 89.4

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

4. Applied rewrites 89.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 85.1

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

7. Applied rewrites 85.1%

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

9. Applied rewrites 85.1%

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

10. Final simplification 85.1%

    \[\leadsto \left(\left(x \cdot x\right) \cdot \left(5 \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right) \]
11. Add Preprocessing

Alternative 11: 83.1% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 89.4

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 89.4

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

4. Applied rewrites 89.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 85.1

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

7. Applied rewrites 85.1%

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

9. Applied rewrites 85.1%

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

10. Final simplification 85.1%

    \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right) \]
11. Add Preprocessing

Alternative 12: 72.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

   9. lower-pow.f64 89.3

      \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

5. Applied rewrites 89.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 89.3%

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

10. Applied rewrites 89.2%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 74.8%

    \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot 5\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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)))