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

Percentage Accurate: 88.8% → 98.6%

Time: 8.1s

Alternatives: 14

Speedup: 3.8×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% 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: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-315}:\\ \;\;\;\;t\_0 - {\left({x}^{4}\right)}^{1.25}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -5e-315)
     (- t_0 (pow (pow x 4.0) 1.25))
     (if (<= t_1 0.0) (* (* 5.0 eps) (pow x 4.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-315) {
		tmp = t_0 - pow(pow(x, 4.0), 1.25);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    if (t_1 <= (-5d-315)) then
        tmp = t_0 - ((x ** 4.0d0) ** 1.25d0)
    else if (t_1 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-315) {
		tmp = t_0 - Math.pow(Math.pow(x, 4.0), 1.25);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	tmp = 0
	if t_1 <= -5e-315:
		tmp = t_0 - math.pow(math.pow(x, 4.0), 1.25)
	elif t_1 <= 0.0:
		tmp = (5.0 * eps) * math.pow(x, 4.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -5e-315)
		tmp = Float64(t_0 - ((x ^ 4.0) ^ 1.25));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	tmp = 0.0;
	if (t_1 <= -5e-315)
		tmp = t_0 - ((x ^ 4.0) ^ 1.25);
	elseif (t_1 <= 0.0)
		tmp = (5.0 * eps) * (x ^ 4.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-315], N[(t$95$0 - N[Power[N[Power[x, 4.0], $MachinePrecision], 1.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -5.0000000023e-315

   1. Initial program 98.4%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. pow-prod-down N/A

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

      9. pow-sqr N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.5

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - {\left({x}^{4}\right)}^{\color{blue}{1.25}} \]

   4. Applied rewrites 98.5%

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{\left({x}^{4}\right)}^{1.25}} \]

   if -5.0000000023e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-315}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-315)
     t_0
     (if (<= t_0 0.0) (* (* 5.0 eps) (pow x 4.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-5d-315)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -5e-315:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (5.0 * eps) * math.pow(x, 4.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-315)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -5e-315)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (5.0 * eps) * (x ^ 4.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-315], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -5.0000000023e-315

   1. Initial program 98.4%

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

   if -5.0000000023e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-315}:\\ \;\;\;\;\left(\left(--1\right) + \frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{\varepsilon}\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-315)
     (*
      (+ (- -1.0) (/ (fma 5.0 x (/ (* -10.0 (* x x)) (- eps))) eps))
      (pow eps 5.0))
     (if (<= t_0 0.0) (* (* 5.0 eps) (pow x 4.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = (-(-1.0) + (fma(5.0, x, ((-10.0 * (x * x)) / -eps)) / eps)) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-315)
		tmp = Float64(Float64(Float64(-(-1.0)) + Float64(fma(5.0, x, Float64(Float64(-10.0 * Float64(x * x)) / Float64(-eps))) / eps)) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-315], N[(N[((--1.0) + N[(N[(5.0 * x + N[(N[(-10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] / (-eps)), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -5.0000000023e-315

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 90.0%

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

   if -5.0000000023e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-315}:\\ \;\;\;\;\left(\left(--1\right) + \frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{\varepsilon}\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-315}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-315)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (if (<= t_0 0.0) (* (* 5.0 eps) (pow x 4.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-315)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-315], 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[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -5.0000000023e-315

   1. Initial program 98.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   if -5.0000000023e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-315} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-315) (not (<= t_0 0.0)))
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (* (* (* (* 5.0 eps) x) x) (* x x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-315) || !(t_0 <= 0.0)) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else {
		tmp = (((5.0 * eps) * x) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-315) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(5.0 * eps) * x) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-315], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

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))) < -5.0000000023e-315 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 94.3

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

   8. Applied rewrites 94.3%

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

   10. Applied rewrites 94.0%

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

   12. Applied rewrites 94.0%

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

   if -5.0000000023e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

   12. Applied rewrites 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-315} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-315}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-315)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (* (* 5.0 eps) (* x x)) x) x)
       (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (((5.0 * eps) * (x * x)) * x) * x;
	} else {
		tmp = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-315)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(5.0 * eps) * Float64(x * x)) * x) * x);
	else
		tmp = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-315], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(5.0 * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $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))) < -5.0000000023e-315

   1. Initial program 98.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 88.5

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

   8. Applied rewrites 88.5%

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

   10. Applied rewrites 87.9%

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

   12. Applied rewrites 88.0%

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

   if -5.0000000023e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

   12. Applied rewrites 99.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 99.4

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

   8. Applied rewrites 99.4%

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

   10. Applied rewrites 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-315}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-315}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-315)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (* (* 5.0 eps) x) x) (* x x))
       (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-315) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (((5.0 * eps) * x) * x) * (x * x);
	} else {
		tmp = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-315)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(5.0 * eps) * x) * x) * Float64(x * x));
	else
		tmp = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-315], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $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))) < -5.0000000023e-315

   1. Initial program 98.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 88.5

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

   8. Applied rewrites 88.5%

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

   10. Applied rewrites 87.9%

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

   12. Applied rewrites 88.0%

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

   if -5.0000000023e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

   12. Applied rewrites 99.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 99.4

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

   8. Applied rewrites 99.4%

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

   10. Applied rewrites 99.3%

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

Alternative 8: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-60} \lor \neg \left(x \leq 4 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon, \frac{-10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{-x}\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-60) (not (<= x 4e-83)))
   (* (fma 5.0 eps (/ (* -10.0 (* eps eps)) (- x))) (pow x 4.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-60) || !(x <= 4e-83)) {
		tmp = fma(5.0, eps, ((-10.0 * (eps * eps)) / -x)) * pow(x, 4.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-60) || !(x <= 4e-83))
		tmp = Float64(fma(5.0, eps, Float64(Float64(-10.0 * Float64(eps * eps)) / Float64(-x))) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-60], N[Not[LessEqual[x, 4e-83]], $MachinePrecision]], N[(N[(5.0 * eps + N[(N[(-10.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000001e-60 or 4.0000000000000001e-83 < x

   1. Initial program 62.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.6%

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

   if -5.0000000000000001e-60 < x < 4.0000000000000001e-83

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-60} \lor \neg \left(x \leq 4 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon, \frac{-10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{-x}\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-60}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-83}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.8e-60)
   (* (* 5.0 eps) (pow x 4.0))
   (if (<= x 4e-83)
     (pow eps 5.0)
     (* (* (* (* (fma (/ eps x) 10.0 5.0) eps) x) x) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.8e-60) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else if (x <= 4e-83) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (((fma((eps / x), 10.0, 5.0) * eps) * x) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.8e-60)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	elseif (x <= 4e-83)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * x) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.8e-60], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-83], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999999e-60

   1. Initial program 49.3%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 96.0%

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

   if -5.7999999999999999e-60 < x < 4.0000000000000001e-83

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 4.0000000000000001e-83 < x

   1. Initial program 70.9%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.9%

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

Alternative 10: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) \cdot 5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-83}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.8e-60)
   (* (* (* (* x x) (* x x)) eps) 5.0)
   (if (<= x 4e-83)
     (pow eps 5.0)
     (* (* (* (* (fma (/ eps x) 10.0 5.0) eps) x) x) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.8e-60) {
		tmp = (((x * x) * (x * x)) * eps) * 5.0;
	} else if (x <= 4e-83) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (((fma((eps / x), 10.0, 5.0) * eps) * x) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.8e-60)
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * eps) * 5.0);
	elseif (x <= 4e-83)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * x) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.8e-60], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], If[LessEqual[x, 4e-83], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999999e-60

   1. Initial program 49.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   if -5.7999999999999999e-60 < x < 4.0000000000000001e-83

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 4.0000000000000001e-83 < x

   1. Initial program 70.9%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.9%

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

Alternative 11: 97.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) \cdot 5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-83}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.8e-60)
   (* (* (* (* x x) (* x x)) eps) 5.0)
   (if (<= x 4e-83)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (* (* (* (* (fma (/ eps x) 10.0 5.0) eps) x) x) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.8e-60) {
		tmp = (((x * x) * (x * x)) * eps) * 5.0;
	} else if (x <= 4e-83) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else {
		tmp = (((fma((eps / x), 10.0, 5.0) * eps) * x) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.8e-60)
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * eps) * 5.0);
	elseif (x <= 4e-83)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * x) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.8e-60], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], If[LessEqual[x, 4e-83], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999999e-60

   1. Initial program 49.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   if -5.7999999999999999e-60 < x < 4.0000000000000001e-83

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   12. Applied rewrites 99.9%

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

   if 4.0000000000000001e-83 < x

   1. Initial program 70.9%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.9%

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

Alternative 12: 82.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 87.6%

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

7. Applied rewrites 87.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 87.5%

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

12. Applied rewrites 87.5%

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

Alternative 13: 82.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 87.6%

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

7. Applied rewrites 87.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 87.5%

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

Alternative 14: 70.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* (* 10.0 eps) (* eps x)) (* x x)))

C

double code(double x, double eps) {
	return ((10.0 * eps) * (eps * x)) * (x * x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((10.0d0 * eps) * (eps * x)) * (x * x)
end function

Java

public static double code(double x, double eps) {
	return ((10.0 * eps) * (eps * x)) * (x * x);
}

Python

def code(x, eps):
	return ((10.0 * eps) * (eps * x)) * (x * x)

Julia

function code(x, eps)
	return Float64(Float64(Float64(10.0 * eps) * Float64(eps * x)) * Float64(x * x))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((10.0 * eps) * (eps * x)) * (x * x);
end

Wolfram

code[x_, eps_] := N[(N[(N[(10.0 * eps), $MachinePrecision] * N[(eps * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 87.6%

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

7. Applied rewrites 87.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 78.7%

    \[\leadsto \left(\left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(\color{blue}{x} \cdot x\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(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)))