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

Percentage Accurate: 88.8% → 99.5%

Time: 11.2s

Alternatives: 11

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -4e-317)
     t_1
     (if (<= t_1 0.0)
       (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
       (- t_0 (* (* x x) (* x (* x x))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -4e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else {
		tmp = t_0 - ((x * x) * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    if (t_1 <= (-4d-317)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x))
    else
        tmp = t_0 - ((x * x) * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -4e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else {
		tmp = t_0 - ((x * x) * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	tmp = 0
	if t_1 <= -4e-317:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x))
	else:
		tmp = t_0 - ((x * x) * (x * (x * x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -4e-317)
		tmp = t_1;
	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 = Float64(t_0 - Float64(Float64(x * x) * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	tmp = 0.0;
	if (t_1 <= -4e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (x ^ 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	else
		tmp = t_0 - ((x * x) * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-317], t$95$1, 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], N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999993e-317

   1. Initial program 93.5%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-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%

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

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

   1. Initial program 99.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. 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 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -4e-317)
     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 <= -4e-317) {
		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 <= (-4d-317)) 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 <= -4e-317) {
		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 <= -4e-317:
		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 <= -4e-317)
		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 <= -4e-317)
		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, -4e-317], 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))) < -3.99999993e-317 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3

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

   4. Applied egg-rr 96.3%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := x \cdot \left(x \cdot x\right)\\ t_3 := t\_0 - \left(x \cdot x\right) \cdot t\_2\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (* x (* x x)))
        (t_3 (- t_0 (* (* x x) t_2))))
   (if (<= t_1 -4e-317) t_3 (if (<= t_1 0.0) (* eps (* 5.0 (* x t_2))) t_3))))

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 = x * (x * x);
	double t_3 = t_0 - ((x * x) * t_2);
	double tmp;
	if (t_1 <= -4e-317) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * (x * t_2));
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = x * (x * x)
    t_3 = t_0 - ((x * x) * t_2)
    if (t_1 <= (-4d-317)) then
        tmp = t_3
    else if (t_1 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x * t_2))
    else
        tmp = t_3
    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 = x * (x * x);
	double t_3 = t_0 - ((x * x) * t_2);
	double tmp;
	if (t_1 <= -4e-317) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * (x * t_2));
	} else {
		tmp = t_3;
	}
	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 = x * (x * x)
	t_3 = t_0 - ((x * x) * t_2)
	tmp = 0
	if t_1 <= -4e-317:
		tmp = t_3
	elif t_1 <= 0.0:
		tmp = eps * (5.0 * (x * t_2))
	else:
		tmp = t_3
	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(x * Float64(x * x))
	t_3 = Float64(t_0 - Float64(Float64(x * x) * t_2))
	tmp = 0.0
	if (t_1 <= -4e-317)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * t_2)));
	else
		tmp = t_3;
	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 = x * (x * x);
	t_3 = t_0 - ((x * x) * t_2);
	tmp = 0.0;
	if (t_1 <= -4e-317)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = eps * (5.0 * (x * t_2));
	else
		tmp = t_3;
	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[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-317], t$95$3, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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))) < -3.99999993e-317 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3

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

   4. Applied egg-rr 96.3%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-317)
     (pow eps 5.0)
     (if (<= t_0 0.0)
       (* eps (* 5.0 (* x (* x (* x x)))))
       (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-317) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-317)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-317], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999993e-317

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 88.4

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

   5. Simplified 88.4%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-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 96.0

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

   5. Simplified 96.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot 10\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\frac{1}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* eps (* eps eps))))
   (if (<= t_0 -4e-317)
     (* t_1 (fma eps (fma 5.0 x eps) (* x (* x 10.0))))
     (if (<= t_0 0.0)
       (* eps (* 5.0 (* x (* x (* x x)))))
       (/ t_1 (/ 1.0 (fma eps (fma 5.0 x eps) (* (* x x) 10.0))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = eps * (eps * eps);
	double tmp;
	if (t_0 <= -4e-317) {
		tmp = t_1 * fma(eps, fma(5.0, x, eps), (x * (x * 10.0)));
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = t_1 / (1.0 / fma(eps, fma(5.0, x, eps), ((x * x) * 10.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -4e-317)
		tmp = Float64(t_1 * fma(eps, fma(5.0, x, eps), Float64(x * Float64(x * 10.0))));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(t_1 / Float64(1.0 / fma(eps, fma(5.0, x, eps), Float64(Float64(x * x) * 10.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-317], N[(t$95$1 * N[(eps * N[(5.0 * x + eps), $MachinePrecision] + N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(1.0 / N[(eps * N[(5.0 * x + eps), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999993e-317

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 88.6%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 88.6

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

   8. Simplified 88.6%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 87.9%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 96.5%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 96.5

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

   8. Simplified 96.5%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 95.5%

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

       1. flip3-+ N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

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

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

   13. Applied egg-rr 95.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot 10\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)}{\frac{1}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot 10\right)\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1
         (* (* eps (* eps eps)) (fma eps (fma 5.0 x eps) (* x (* x 10.0))))))
   (if (<= t_0 -4e-317)
     t_1
     (if (<= t_0 0.0) (* eps (* 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 = (eps * (eps * eps)) * fma(eps, fma(5.0, x, eps), (x * (x * 10.0)));
	double tmp;
	if (t_0 <= -4e-317) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (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(Float64(eps * Float64(eps * eps)) * fma(eps, fma(5.0, x, eps), Float64(x * Float64(x * 10.0))))
	tmp = 0.0
	if (t_0 <= -4e-317)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * 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[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(5.0 * x + eps), $MachinePrecision] + N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-317], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $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))) < -3.99999993e-317 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 92.6%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 92.6

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

   8. Simplified 92.6%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 91.8%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot 10\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* eps (* eps eps))))
   (if (<= t_0 -4e-317)
     (* (* eps eps) t_1)
     (if (<= t_0 0.0)
       (* eps (* 5.0 (* x (* x (* x x)))))
       (* (fma 5.0 x eps) (* 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 = eps * (eps * eps);
	double tmp;
	if (t_0 <= -4e-317) {
		tmp = (eps * eps) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = fma(5.0, x, eps) * (eps * t_1);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -4e-317)
		tmp = Float64(Float64(eps * eps) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(fma(5.0, x, eps) * Float64(eps * 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[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-317], N[(N[(eps * eps), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999993e-317

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 88.6%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

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

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 87.6

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

   8. Simplified 87.6%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 87.7

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

   10. Applied egg-rr 87.7%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 96.5%

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

   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. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. +-commutative N/A

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

      18. accelerator-lowering-fma.f64 95.4

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

   8. Simplified 95.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-317}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* eps (* eps eps))))
   (if (<= t_0 -4e-317)
     (* (* eps eps) t_1)
     (if (<= t_0 0.0)
       (* eps (* 5.0 (* x (* x (* x x)))))
       (* eps (* 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 = eps * (eps * eps);
	double tmp;
	if (t_0 <= -4e-317) {
		tmp = (eps * eps) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = eps * (eps * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = eps * (eps * eps)
    if (t_0 <= (-4d-317)) then
        tmp = (eps * eps) * t_1
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x * (x * (x * x))))
    else
        tmp = eps * (eps * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double t_1 = eps * (eps * eps);
	double tmp;
	if (t_0 <= -4e-317) {
		tmp = (eps * eps) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = eps * (eps * t_1);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = eps * (eps * eps)
	tmp = 0
	if t_0 <= -4e-317:
		tmp = (eps * eps) * t_1
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * (x * (x * (x * x))))
	else:
		tmp = eps * (eps * t_1)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -4e-317)
		tmp = Float64(Float64(eps * eps) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(eps * Float64(eps * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = eps * (eps * eps);
	tmp = 0.0;
	if (t_0 <= -4e-317)
		tmp = (eps * eps) * t_1;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x * (x * (x * x))));
	else
		tmp = eps * (eps * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-317], N[(N[(eps * eps), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999993e-317

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 88.6%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

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

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 87.6

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

   8. Simplified 87.6%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 87.7

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

   10. Applied egg-rr 87.7%

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

   if -3.99999993e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 96.5%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

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

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 93.9

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

   8. Simplified 93.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-317}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -5.3 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot \mathsf{fma}\left(\varepsilon, \frac{10}{x}, 5\right)\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(5, t\_0, \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -5.3e-43)
     (* eps (* t_0 (fma eps (/ 10.0 x) 5.0)))
     (if (<= x 1.2e-66)
       (pow eps 5.0)
       (* eps (fma 5.0 t_0 (* eps (* (+ x eps) (* (* x x) 10.0)))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -5.3e-43) {
		tmp = eps * (t_0 * fma(eps, (10.0 / x), 5.0));
	} else if (x <= 1.2e-66) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * fma(5.0, t_0, (eps * ((x + eps) * ((x * x) * 10.0))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -5.3e-43)
		tmp = Float64(eps * Float64(t_0 * fma(eps, Float64(10.0 / x), 5.0)));
	elseif (x <= 1.2e-66)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * fma(5.0, t_0, Float64(eps * Float64(Float64(x + eps) * Float64(Float64(x * x) * 10.0)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.3e-43], N[(eps * N[(t$95$0 * N[(eps * N[(10.0 / x), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-66], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * t$95$0 + N[(eps * N[(N[(x + eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3000000000000003e-43

   1. Initial program 13.9%

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

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

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

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

      15. /-lowering-/.f64 99.5

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

   8. Simplified 99.5%

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

   if -5.3000000000000003e-43 < x < 1.20000000000000013e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.20000000000000013e-66 < x

   1. Initial program 58.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 89.3%

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

      1. *-commutative N/A

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

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

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

   7. Applied egg-rr 89.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \frac{10}{x}, 5\right)\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(5, x \cdot \left(x \cdot \left(x \cdot x\right)\right), \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.8% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in x around 0

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

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

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

5. Simplified 86.6%

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

6. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

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

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   5. metadata-eval N/A

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

   6. pow-plus N/A

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

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

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 86.2

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

8. Simplified 86.2%

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

9. Final simplification 86.2%

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

Alternative 11: 87.8% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in x around 0

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

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

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

5. Simplified 86.6%

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

6. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

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

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   5. metadata-eval N/A

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

   6. pow-plus N/A

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

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

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 86.2

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

8. Simplified 86.2%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-lowering-*.f64 86.2

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

10. Applied egg-rr 86.2%

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

Reproduce

herbie shell --seed 2024198 
(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)))