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

Percentage Accurate: 87.8% → 99.5%

Time: 7.8s

Alternatives: 14

Speedup: 0.5×

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: 87.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-320)
     t_0
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) t_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 <= -5e-320) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_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 <= (-5d-320)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = t_0
    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 <= -5e-320) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_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 <= -5e-320:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = t_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 <= -5e-320)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = t_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 <= -5e-320)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = t_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, -5e-320], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := {\varepsilon}^{5} \cdot \left(1 + \frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1 (* (pow eps 5.0) (+ 1.0 (/ (* (fma 10.0 (/ x eps) 5.0) x) eps)))))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = pow(eps, 5.0) * (1.0 + ((fma(10.0, (x / eps), 5.0) * x) / eps));
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64((eps ^ 5.0) * Float64(1.0 + Float64(Float64(fma(10.0, Float64(x / eps), 5.0) * x) / eps)))
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(N[(N[(10.0 * N[(x / eps), $MachinePrecision] + 5.0), $MachinePrecision] * x), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 95.9%

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

   6. Taylor expanded in eps around inf

      \[\leadsto \left(\frac{5 \cdot x + 10 \cdot \frac{{x}^{2}}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.9%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + \frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + \frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, x, {\varepsilon}^{5}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-320)
     (fma (* (* (* eps eps) (* eps eps)) 5.0) x (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* (pow x 4.0) 5.0) eps)
       (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (pow eps 3.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = fma((((eps * eps) * (eps * eps)) * 5.0), x, pow(eps, 5.0));
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * pow(eps, 3.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = fma(Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), x, (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * (eps ^ 3.0));
	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, -5e-320], N[(N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * x + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $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))) < -4.99994e-320

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 96.6

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.6%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   10. Applied rewrites 94.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, x, {\varepsilon}^{5}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, x, {\varepsilon}^{5}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(10, x \cdot x, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot {\varepsilon}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-320)
     (fma (* (* (* eps eps) (* eps eps)) 5.0) x (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* (pow x 4.0) 5.0) eps)
       (* (fma 10.0 (* x x) (* (fma 5.0 x eps) eps)) (pow eps 3.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = fma((((eps * eps) * (eps * eps)) * 5.0), x, pow(eps, 5.0));
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = fma(10.0, (x * x), (fma(5.0, x, eps) * eps)) * pow(eps, 3.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = fma(Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), x, (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = Float64(fma(10.0, Float64(x * x), Float64(fma(5.0, x, eps) * eps)) * (eps ^ 3.0));
	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, -5e-320], N[(N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * x + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], 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]]]]

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))) < -4.99994e-320

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 96.6

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.6%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, x, {\varepsilon}^{5}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(10, x \cdot x, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot {\varepsilon}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, x, {\varepsilon}^{5}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-320)
     (fma (* (* (* eps eps) (* eps eps)) 5.0) x (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* (pow x 4.0) 5.0) eps)
       (* (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) eps) 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 <= -5e-320) {
		tmp = fma((((eps * eps) * (eps * eps)) * 5.0), x, pow(eps, 5.0));
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = ((fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * eps) * 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 <= -5e-320)
		tmp = fma(Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), x, (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = Float64(Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * 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, -5e-320], N[(N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * x + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $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))) < -4.99994e-320

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 96.6

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.6%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   10. Applied rewrites 94.5%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 94.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, x, {\varepsilon}^{5}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-320)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* (pow x 4.0) 5.0) eps)
       (* (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) eps) 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 <= -5e-320) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = ((fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * eps) * 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 <= -5e-320)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = Float64(Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * 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, -5e-320], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $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))) < -4.99994e-320

   1. Initial program 98.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.5

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

   5. Applied rewrites 96.5%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   10. Applied rewrites 94.5%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 94.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1
         (* (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) eps) eps)))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * eps) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * eps)
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 95.5%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(\left(\varepsilon + x\right) \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1
         (* (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) eps) eps)))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0)
       (* (* (* (fma (* 5.0 x) x (* (* (+ eps x) eps) 10.0)) x) x) eps)
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * eps) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((fma((5.0 * x), x, (((eps + x) * eps) * 10.0)) * x) * x) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * eps)
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(Float64(eps + x) * eps) * 10.0)) * x) * x) * eps);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(N[(eps + x), $MachinePrecision] * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 95.5%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(\left(\varepsilon + x\right) \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1
         (* (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) eps) eps)))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0) (* (* (* (* x x) 5.0) eps) (* x x)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * eps) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * eps)
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 95.5%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

       \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1 (* (* (* (fma 5.0 x eps) eps) (* eps eps)) eps)))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0) (* (* (* (* x x) 5.0) eps) (* x x)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((fma(5.0, x, eps) * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(fma(5.0, x, eps) * eps) * Float64(eps * eps)) * eps)
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 94.8%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

       \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1 (* (* (* eps eps) (* eps eps)) eps)))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0) (* (* (* (* x x) 5.0) eps) (* x x)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((eps + x) ** 5.0d0) - (x ** 5.0d0)
    t_1 = ((eps * eps) * (eps * eps)) * eps
    if (t_0 <= (-5d-320)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (((x * x) * 5.0d0) * eps) * (x * x)
    else
        tmp = t_1
    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 t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((eps + x), 5.0) - math.pow(x, 5.0)
	t_1 = ((eps * eps) * (eps * eps)) * eps
	tmp = 0
	if t_0 <= -5e-320:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (((x * x) * 5.0) * eps) * (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps)
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((eps + x) ^ 5.0) - (x ^ 5.0);
	t_1 = ((eps * eps) * (eps * eps)) * eps;
	tmp = 0.0;
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 93.7%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

       \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1 (* (* (* eps eps) (* eps eps)) eps)))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0) (* (* (* 5.0 eps) (* x x)) (* x x)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((5.0 * eps) * (x * x)) * (x * x);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((eps + x) ** 5.0d0) - (x ** 5.0d0)
    t_1 = ((eps * eps) * (eps * eps)) * eps
    if (t_0 <= (-5d-320)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = ((5.0d0 * eps) * (x * x)) * (x * x)
    else
        tmp = t_1
    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 t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((5.0 * eps) * (x * x)) * (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((eps + x), 5.0) - math.pow(x, 5.0)
	t_1 = ((eps * eps) * (eps * eps)) * eps
	tmp = 0
	if t_0 <= -5e-320:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = ((5.0 * eps) * (x * x)) * (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps)
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(5.0 * eps) * Float64(x * x)) * Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((eps + x) ^ 5.0) - (x ^ 5.0);
	t_1 = ((eps * eps) * (eps * eps)) * eps;
	tmp = 0.0;
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = ((5.0 * eps) * (x * x)) * (x * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(5.0 * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 93.7%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1 (* (* (* eps eps) (* eps eps)) eps)))
   (if (<= t_0 -5e-320)
     t_1
     (if (<= t_0 0.0) (* (* (* (* (* x x) 5.0) eps) x) x) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((((x * x) * 5.0) * eps) * x) * x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((eps + x) ** 5.0d0) - (x ** 5.0d0)
    t_1 = ((eps * eps) * (eps * eps)) * eps
    if (t_0 <= (-5d-320)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = ((((x * x) * 5.0d0) * eps) * x) * x
    else
        tmp = t_1
    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 t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -5e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((((x * x) * 5.0) * eps) * x) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((eps + x), 5.0) - math.pow(x, 5.0)
	t_1 = ((eps * eps) * (eps * eps)) * eps
	tmp = 0
	if t_0 <= -5e-320:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = ((((x * x) * 5.0) * eps) * x) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps)
	tmp = 0.0
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * x) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((eps + x) ^ 5.0) - (x ^ 5.0);
	t_1 = ((eps * eps) * (eps * eps)) * eps;
	tmp = 0.0;
	if (t_0 <= -5e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = ((((x * x) * 5.0) * eps) * x) * x;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 93.7%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

       \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 86.9

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

5. Applied rewrites 86.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 86.9%

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

10. Applied rewrites 86.9%

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

11. Taylor expanded in eps around inf

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

13. Applied rewrites 86.5%

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

Reproduce

herbie shell --seed 2024276 
(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)))