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

Percentage Accurate: 88.4% → 99.4%

Time: 10.6s

Alternatives: 13

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* x x) (* x (* x x))))))
   (if (<= t_1 -2e-305)
     t_2
     (if (<= t_1 0.0)
       (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
       t_2))))

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 t_2 = t_0 - ((x * x) * (x * (x * x)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - ((x * x) * (x * (x * x)))
    if (t_1 <= (-2d-305)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x))
    else
        tmp = t_2
    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 t_2 = t_0 - ((x * x) * (x * (x * x)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - ((x * x) * (x * (x * x)))
	tmp = 0
	if t_1 <= -2e-305:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(x * x) * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - ((x * x) * (x * (x * x)));
	tmp = 0.0;
	if (t_1 <= -2e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = (x ^ 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-305], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

      1. --lowering--.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 98.9

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

   4. Applied egg-rr 98.9%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   5. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* x x) (* x (* x x))))))
   (if (<= t_1 -2e-305)
     t_2
     (if (<= t_1 0.0) (* eps (* 5.0 (pow x 4.0))) t_2))))

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 t_2 = t_0 - ((x * x) * (x * (x * x)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - ((x * x) * (x * (x * x)))
    if (t_1 <= (-2d-305)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_2
    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 t_2 = t_0 - ((x * x) * (x * (x * x)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - ((x * x) * (x * (x * x)))
	tmp = 0
	if t_1 <= -2e-305:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(x * x) * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - ((x * x) * (x * (x * x)));
	tmp = 0.0;
	if (t_1 <= -2e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-305], t$95$2, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

      1. --lowering--.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 98.9

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

   4. Applied egg-rr 98.9%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-lowering-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 99.9

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

   5. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := {\varepsilon}^{5} \cdot \frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (pow eps 5.0) (/ (fma 5.0 x eps) eps))))
   (if (<= t_0 -2e-289)
     t_1
     (if (<= t_0 0.0)
       (/
        eps
        (/
         1.0
         (fma eps (* (+ x eps) (* (* x x) 10.0)) (* 5.0 (* x (* x (* x x)))))))
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = pow(eps, 5.0) * (fma(5.0, x, eps) / eps);
	double tmp;
	if (t_0 <= -2e-289) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps / (1.0 / fma(eps, ((x + eps) * ((x * x) * 10.0)), (5.0 * (x * (x * (x * x))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64((eps ^ 5.0) * Float64(fma(5.0, x, eps) / eps))
	tmp = 0.0
	if (t_0 <= -2e-289)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps / Float64(1.0 / fma(eps, Float64(Float64(x + eps) * Float64(Float64(x * x) * 10.0)), Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Power[eps, 5.0], $MachinePrecision] * N[(N[(5.0 * x + eps), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-289], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps / N[(1.0 / N[(eps * N[(N[(x + eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2e-289 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]

      2. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \frac{\color{blue}{5 \cdot x + \varepsilon}}{\varepsilon} \]

      3. accelerator-lowering-fma.f64 96.0

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

   8. Simplified 96.0%

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

   if -2e-289 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\frac{1}{\frac{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) \cdot \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) + \left(\left(5 \cdot {x}^{4}\right) \cdot \left(5 \cdot {x}^{4}\right) - \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) \cdot \left(5 \cdot {x}^{4}\right)\right)}{{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right)}^{3} + {\left(5 \cdot {x}^{4}\right)}^{3}}}} \]

      3. un-div-inv N/A

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

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(\left(x \cdot x\right) \cdot 10\right) \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-289}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (pow eps 4.0))))
   (if (<= t_0 -2e-289)
     t_1
     (if (<= t_0 0.0)
       (/
        eps
        (/
         1.0
         (fma eps (* (+ x eps) (* (* x x) 10.0)) (* 5.0 (* x (* x (* x x)))))))
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = fma(5.0, x, eps) * pow(eps, 4.0);
	double tmp;
	if (t_0 <= -2e-289) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps / (1.0 / fma(eps, ((x + eps) * ((x * x) * 10.0)), (5.0 * (x * (x * (x * x))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(fma(5.0, x, eps) * (eps ^ 4.0))
	tmp = 0.0
	if (t_0 <= -2e-289)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps / Float64(1.0 / fma(eps, Float64(Float64(x + eps) * Float64(Float64(x * x) * 10.0)), Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(5.0 * x + eps), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-289], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps / N[(1.0 / N[(eps * N[(N[(x + eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2e-289 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. pow-lowering-pow.f64 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow-lowering-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 95.6

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

   8. Simplified 95.6%

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

   if -2e-289 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\frac{1}{\frac{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) \cdot \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) + \left(\left(5 \cdot {x}^{4}\right) \cdot \left(5 \cdot {x}^{4}\right) - \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) \cdot \left(5 \cdot {x}^{4}\right)\right)}{{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right)}^{3} + {\left(5 \cdot {x}^{4}\right)}^{3}}}} \]

      3. un-div-inv N/A

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

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(\left(x \cdot x\right) \cdot 10\right) \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))))
   (if (<= t_0 -2e-289)
     t_1
     (if (<= t_0 0.0)
       (/
        eps
        (/
         1.0
         (fma eps (* (+ x eps) (* (* x x) 10.0)) (* 5.0 (* x (* x (* x x)))))))
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	double tmp;
	if (t_0 <= -2e-289) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps / (1.0 / fma(eps, ((x + eps) * ((x * x) * 10.0)), (5.0 * (x * (x * (x * x))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-289)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps / Float64(1.0 / fma(eps, Float64(Float64(x + eps) * Float64(Float64(x * x) * 10.0)), Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-289], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps / N[(1.0 / N[(eps * N[(N[(x + eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2e-289 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]

      2. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \frac{\color{blue}{5 \cdot x + \varepsilon}}{\varepsilon} \]

      3. accelerator-lowering-fma.f64 96.0

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

   8. Simplified 96.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{5 \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5}} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 95.2

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

   10. Applied egg-rr 95.2%

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

   if -2e-289 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\frac{1}{\frac{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) \cdot \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) + \left(\left(5 \cdot {x}^{4}\right) \cdot \left(5 \cdot {x}^{4}\right) - \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) \cdot \left(5 \cdot {x}^{4}\right)\right)}{{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right)}^{3} + {\left(5 \cdot {x}^{4}\right)}^{3}}}} \]

      3. un-div-inv N/A

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

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(\left(x \cdot x\right) \cdot 10\right) \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(\varepsilon, \left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(5, \varepsilon \cdot \left(x \cdot x\right), 10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))))
   (if (<= t_0 -2e-289)
     t_1
     (if (<= t_0 0.0)
       (* x (* x (fma 5.0 (* eps (* x x)) (* 10.0 (* eps (* eps (+ x eps)))))))
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	double tmp;
	if (t_0 <= -2e-289) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = x * (x * fma(5.0, (eps * (x * x)), (10.0 * (eps * (eps * (x + eps))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-289)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(x * fma(5.0, Float64(eps * Float64(x * x)), Float64(10.0 * Float64(eps * Float64(eps * Float64(x + eps)))))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-289], t$95$1, If[LessEqual[t$95$0, 0.0], N[(x * N[(x * N[(5.0 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[(eps * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2e-289 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]

      2. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \frac{\color{blue}{5 \cdot x + \varepsilon}}{\varepsilon} \]

      3. accelerator-lowering-fma.f64 96.0

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

   8. Simplified 96.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{5 \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5}} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 95.2

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

   10. Applied egg-rr 95.2%

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

   if -2e-289 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-out N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. +-lowering-+.f64 99.4

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. associate-+l+ N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(5, \varepsilon \cdot \left(x \cdot x\right), 10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))))
   (if (<= t_0 -2e-289)
     t_1
     (if (<= t_0 0.0)
       (* eps (* (* x (* x x)) (fma eps 10.0 (* x 5.0))))
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	double tmp;
	if (t_0 <= -2e-289) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * ((x * (x * x)) * fma(eps, 10.0, (x * 5.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-289)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(Float64(x * Float64(x * x)) * fma(eps, 10.0, Float64(x * 5.0))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-289], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2e-289 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]

      2. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \frac{\color{blue}{5 \cdot x + \varepsilon}}{\varepsilon} \]

      3. accelerator-lowering-fma.f64 96.0

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

   8. Simplified 96.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{5 \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5}} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 95.2

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

   10. Applied egg-rr 95.2%

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

   if -2e-289 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0

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

      1. metadata-eval N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(3 + 1\right)}} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. *-lowering-*.f64 99.4

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

   8. Simplified 99.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot 5\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0) (* (* x 5.0) (* x (* eps (* x x)))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * 5.0) * (x * (eps * (x * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * 5.0) * Float64(x * Float64(eps * Float64(x * x))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(x * 5.0), $MachinePrecision] * N[(x * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\frac{\varepsilon + 5 \cdot x}{\varepsilon}} \]

      2. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \frac{\color{blue}{5 \cdot x + \varepsilon}}{\varepsilon} \]

      3. accelerator-lowering-fma.f64 94.3

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

   8. Simplified 94.3%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{5 \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5}} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 93.5

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

   10. Applied egg-rr 93.5%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-lowering-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 99.9

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

   5. Simplified 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

      2. pow-plus N/A

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

      3. cube-unmult N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 99.9

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot 5\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-lowering-*.f64 N/A

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

   7. distribute-lft1-in N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. pow-lowering-pow.f64 81.6

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

5. Simplified 81.6%

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

   1. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

   2. pow-plus N/A

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

   3. cube-unmult N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 81.5

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

7. Applied egg-rr 81.5%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 81.5

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

9. Applied egg-rr 81.5%

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

10. Final simplification 81.5%

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

Alternative 10: 82.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-lowering-*.f64 N/A

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

   7. distribute-lft1-in N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. pow-lowering-pow.f64 81.6

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

5. Simplified 81.6%

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

   1. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

   2. pow-plus N/A

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

   3. cube-unmult N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 81.5

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

7. Applied egg-rr 81.5%

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

8. Final simplification 81.5%

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

Alternative 11: 82.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-lowering-*.f64 N/A

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

   7. distribute-lft1-in N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. pow-lowering-pow.f64 81.6

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

5. Simplified 81.6%

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

   1. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

   2. pow-plus N/A

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

   3. cube-unmult N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 81.5

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

7. Applied egg-rr 81.5%

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

8. Final simplification 81.5%

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

Alternative 12: 82.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

5. Simplified 81.8%

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

6. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

   2. pow-plus N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. unpow3 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 81.5

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

8. Simplified 81.5%

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

Alternative 13: 71.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

5. Simplified 81.9%

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

6. Taylor expanded in eps around inf

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

   1. *-lowering-*.f64 N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 68.6

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

8. Simplified 68.6%

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

9. Final simplification 68.6%

   \[\leadsto 10 \cdot \left(\varepsilon \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right)\right)\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))