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

Percentage Accurate: 88.3% → 97.2%

Time: 10.4s

Alternatives: 12

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% 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: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-318}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-262}:\\ \;\;\;\;{x}^{3} \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-193}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\frac{{\left(x + \varepsilon\right)}^{10} - {x}^{10}}{t\_0 + {x}^{5}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

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 -2e-318)
     t_1
     (if (<= t_1 0.0)
       (* eps (* 5.0 (pow x 4.0)))
       (if (<= t_1 1e-262)
         (* (pow x 3.0) (+ (* 5.0 (* x eps)) (* 10.0 (pow eps 2.0))))
         (if (<= t_1 5e-193)
           (* (pow eps 4.0) (* x (+ 5.0 (/ eps x))))
           (if (<= t_1 1e-145)
             t_1
             (if (<= t_1 5e-108)
               (/ (- (pow (+ x eps) 10.0) (pow x 10.0)) (+ t_0 (pow x 5.0)))
               (if (<= t_1 5e-19)
                 (* (pow eps 4.0) (+ eps (* x 5.0)))
                 (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps 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 <= -2e-318) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (t_1 <= 1e-262) {
		tmp = pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * pow(eps, 2.0)));
	} else if (t_1 <= 5e-193) {
		tmp = pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	} else if (t_1 <= 1e-145) {
		tmp = t_1;
	} else if (t_1 <= 5e-108) {
		tmp = (pow((x + eps), 10.0) - pow(x, 10.0)) / (t_0 + pow(x, 5.0));
	} else if (t_1 <= 5e-19) {
		tmp = pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / 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 <= (-2d-318)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (t_1 <= 1d-262) then
        tmp = (x ** 3.0d0) * ((5.0d0 * (x * eps)) + (10.0d0 * (eps ** 2.0d0)))
    else if (t_1 <= 5d-193) then
        tmp = (eps ** 4.0d0) * (x * (5.0d0 + (eps / x)))
    else if (t_1 <= 1d-145) then
        tmp = t_1
    else if (t_1 <= 5d-108) then
        tmp = (((x + eps) ** 10.0d0) - (x ** 10.0d0)) / (t_0 + (x ** 5.0d0))
    else if (t_1 <= 5d-19) then
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    else
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / 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 <= -2e-318) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (t_1 <= 1e-262) {
		tmp = Math.pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * Math.pow(eps, 2.0)));
	} else if (t_1 <= 5e-193) {
		tmp = Math.pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	} else if (t_1 <= 1e-145) {
		tmp = t_1;
	} else if (t_1 <= 5e-108) {
		tmp = (Math.pow((x + eps), 10.0) - Math.pow(x, 10.0)) / (t_0 + Math.pow(x, 5.0));
	} else if (t_1 <= 5e-19) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / 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 <= -2e-318:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif t_1 <= 1e-262:
		tmp = math.pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * math.pow(eps, 2.0)))
	elif t_1 <= 5e-193:
		tmp = math.pow(eps, 4.0) * (x * (5.0 + (eps / x)))
	elif t_1 <= 1e-145:
		tmp = t_1
	elif t_1 <= 5e-108:
		tmp = (math.pow((x + eps), 10.0) - math.pow(x, 10.0)) / (t_0 + math.pow(x, 5.0))
	elif t_1 <= 5e-19:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / 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 <= -2e-318)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (t_1 <= 1e-262)
		tmp = Float64((x ^ 3.0) * Float64(Float64(5.0 * Float64(x * eps)) + Float64(10.0 * (eps ^ 2.0))));
	elseif (t_1 <= 5e-193)
		tmp = Float64((eps ^ 4.0) * Float64(x * Float64(5.0 + Float64(eps / x))));
	elseif (t_1 <= 1e-145)
		tmp = t_1;
	elseif (t_1 <= 5e-108)
		tmp = Float64(Float64((Float64(x + eps) ^ 10.0) - (x ^ 10.0)) / Float64(t_0 + (x ^ 5.0)));
	elseif (t_1 <= 5e-19)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / 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 <= -2e-318)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (t_1 <= 1e-262)
		tmp = (x ^ 3.0) * ((5.0 * (x * eps)) + (10.0 * (eps ^ 2.0)));
	elseif (t_1 <= 5e-193)
		tmp = (eps ^ 4.0) * (x * (5.0 + (eps / x)));
	elseif (t_1 <= 1e-145)
		tmp = t_1;
	elseif (t_1 <= 5e-108)
		tmp = (((x + eps) ^ 10.0) - (x ^ 10.0)) / (t_0 + (x ^ 5.0));
	elseif (t_1 <= 5e-19)
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / 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, -2e-318], t$95$1, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-262], N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(5.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-193], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-145], t$95$1, If[LessEqual[t$95$1, 5e-108], N[(N[(N[Power[N[(x + eps), $MachinePrecision], 10.0], $MachinePrecision] - N[Power[x, 10.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-19], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2.0000024e-318 or 5.0000000000000005e-193 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 9.99999999999999915e-146

   1. Initial program 99.8%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   5. Simplified 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 1.00000000000000001e-262

   1. Initial program 43.9%

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

   3. Taylor expanded in x around -inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-+r+ 98.4%

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

      3. mul-1-neg 98.4%

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

      4. unsub-neg 98.4%

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

      5. distribute-rgt1-in 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-commutative 98.4%

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

   5. Simplified 98.4%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0 100.0%

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

   if 1.00000000000000001e-262 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5.0000000000000005e-193

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-define 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 99.2%

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

   7. Taylor expanded in x around inf 100.0%

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

   if 9.99999999999999915e-146 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5e-108

   1. Initial program 99.2%

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

      1. flip-- 100.0%

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

      2. pow-prod-up 100.0%

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

      3. metadata-eval 100.0%

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

      4. pow-prod-up 100.0%

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

      5. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 5e-108 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5.0000000000000004e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-define 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

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

   if 5.0000000000000004e-19 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 42.3%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. distribute-rgt1-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(\varepsilon \cdot \left(5 + \frac{\varepsilon}{x} \cdot 10\right)\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 10^{-262}:\\ \;\;\;\;{x}^{3} \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-193}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 10^{-145}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\frac{{\left(x + \varepsilon\right)}^{10} - {x}^{10}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-318}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-262}:\\ \;\;\;\;{x}^{3} \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-193}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-318)
     t_0
     (if (<= t_0 0.0)
       (* eps (* 5.0 (pow x 4.0)))
       (if (<= t_0 1e-262)
         (* (pow x 3.0) (+ (* 5.0 (* x eps)) (* 10.0 (pow eps 2.0))))
         (if (<= t_0 5e-193)
           (* (pow eps 4.0) (* x (+ 5.0 (/ eps x))))
           (if (<= t_0 5e-108)
             t_0
             (if (<= t_0 5e-19)
               (* (pow eps 4.0) (+ eps (* x 5.0)))
               (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-318) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (t_0 <= 1e-262) {
		tmp = pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * pow(eps, 2.0)));
	} else if (t_0 <= 5e-193) {
		tmp = pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	} else if (t_0 <= 5e-108) {
		tmp = t_0;
	} else if (t_0 <= 5e-19) {
		tmp = pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / 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) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-2d-318)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (t_0 <= 1d-262) then
        tmp = (x ** 3.0d0) * ((5.0d0 * (x * eps)) + (10.0d0 * (eps ** 2.0d0)))
    else if (t_0 <= 5d-193) then
        tmp = (eps ** 4.0d0) * (x * (5.0d0 + (eps / x)))
    else if (t_0 <= 5d-108) then
        tmp = t_0
    else if (t_0 <= 5d-19) then
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    else
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / 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) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-318) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (t_0 <= 1e-262) {
		tmp = Math.pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * Math.pow(eps, 2.0)));
	} else if (t_0 <= 5e-193) {
		tmp = Math.pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	} else if (t_0 <= 5e-108) {
		tmp = t_0;
	} else if (t_0 <= 5e-19) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -2e-318:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif t_0 <= 1e-262:
		tmp = math.pow(x, 3.0) * ((5.0 * (x * eps)) + (10.0 * math.pow(eps, 2.0)))
	elif t_0 <= 5e-193:
		tmp = math.pow(eps, 4.0) * (x * (5.0 + (eps / x)))
	elif t_0 <= 5e-108:
		tmp = t_0
	elif t_0 <= 5e-19:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-318)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (t_0 <= 1e-262)
		tmp = Float64((x ^ 3.0) * Float64(Float64(5.0 * Float64(x * eps)) + Float64(10.0 * (eps ^ 2.0))));
	elseif (t_0 <= 5e-193)
		tmp = Float64((eps ^ 4.0) * Float64(x * Float64(5.0 + Float64(eps / x))));
	elseif (t_0 <= 5e-108)
		tmp = t_0;
	elseif (t_0 <= 5e-19)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -2e-318)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (t_0 <= 1e-262)
		tmp = (x ^ 3.0) * ((5.0 * (x * eps)) + (10.0 * (eps ^ 2.0)));
	elseif (t_0 <= 5e-193)
		tmp = (eps ^ 4.0) * (x * (5.0 + (eps / x)));
	elseif (t_0 <= 5e-108)
		tmp = t_0;
	elseif (t_0 <= 5e-19)
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-318], t$95$0, If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-262], N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(5.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-193], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-108], t$95$0, If[LessEqual[t$95$0, 5e-19], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2.0000024e-318 or 5.0000000000000005e-193 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5e-108

   1. Initial program 99.8%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   5. Simplified 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 1.00000000000000001e-262

   1. Initial program 43.9%

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

   3. Taylor expanded in x around -inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-+r+ 98.4%

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

      3. mul-1-neg 98.4%

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

      4. unsub-neg 98.4%

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

      5. distribute-rgt1-in 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-commutative 98.4%

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

   5. Simplified 98.4%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0 100.0%

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

   if 1.00000000000000001e-262 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5.0000000000000005e-193

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-define 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 99.2%

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

   7. Taylor expanded in x around inf 100.0%

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

   if 5e-108 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5.0000000000000004e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-define 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

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

   if 5.0000000000000004e-19 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 42.3%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. distribute-rgt1-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(\varepsilon \cdot \left(5 + \frac{\varepsilon}{x} \cdot 10\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 10^{-262}:\\ \;\;\;\;{x}^{3} \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-193}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-108}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := {x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-318}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 10^{-262}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-193}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))))
   (if (<= t_0 -2e-318)
     t_0
     (if (<= t_0 1e-262)
       t_1
       (if (<= t_0 5e-193)
         (* (pow eps 4.0) (* x (+ 5.0 (/ eps x))))
         (if (<= t_0 5e-108)
           t_0
           (if (<= t_0 5e-19) (* (pow eps 4.0) (+ eps (* x 5.0))) t_1)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	double tmp;
	if (t_0 <= -2e-318) {
		tmp = t_0;
	} else if (t_0 <= 1e-262) {
		tmp = t_1;
	} else if (t_0 <= 5e-193) {
		tmp = pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	} else if (t_0 <= 5e-108) {
		tmp = t_0;
	} else if (t_0 <= 5e-19) {
		tmp = pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / x))))
    if (t_0 <= (-2d-318)) then
        tmp = t_0
    else if (t_0 <= 1d-262) then
        tmp = t_1
    else if (t_0 <= 5d-193) then
        tmp = (eps ** 4.0d0) * (x * (5.0d0 + (eps / x)))
    else if (t_0 <= 5d-108) then
        tmp = t_0
    else if (t_0 <= 5d-19) then
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    else
        tmp = 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 = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	double tmp;
	if (t_0 <= -2e-318) {
		tmp = t_0;
	} else if (t_0 <= 1e-262) {
		tmp = t_1;
	} else if (t_0 <= 5e-193) {
		tmp = Math.pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	} else if (t_0 <= 5e-108) {
		tmp = t_0;
	} else if (t_0 <= 5e-19) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	tmp = 0
	if t_0 <= -2e-318:
		tmp = t_0
	elif t_0 <= 1e-262:
		tmp = t_1
	elif t_0 <= 5e-193:
		tmp = math.pow(eps, 4.0) * (x * (5.0 + (eps / x)))
	elif t_0 <= 5e-108:
		tmp = t_0
	elif t_0 <= 5e-19:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))))
	tmp = 0.0
	if (t_0 <= -2e-318)
		tmp = t_0;
	elseif (t_0 <= 1e-262)
		tmp = t_1;
	elseif (t_0 <= 5e-193)
		tmp = Float64((eps ^ 4.0) * Float64(x * Float64(5.0 + Float64(eps / x))));
	elseif (t_0 <= 5e-108)
		tmp = t_0;
	elseif (t_0 <= 5e-19)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	tmp = 0.0;
	if (t_0 <= -2e-318)
		tmp = t_0;
	elseif (t_0 <= 1e-262)
		tmp = t_1;
	elseif (t_0 <= 5e-193)
		tmp = (eps ^ 4.0) * (x * (5.0 + (eps / x)));
	elseif (t_0 <= 5e-108)
		tmp = t_0;
	elseif (t_0 <= 5e-19)
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = 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[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-318], t$95$0, If[LessEqual[t$95$0, 1e-262], t$95$1, If[LessEqual[t$95$0, 5e-193], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-108], t$95$0, If[LessEqual[t$95$0, 5e-19], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2.0000024e-318 or 5.0000000000000005e-193 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5e-108

   1. Initial program 99.8%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 1.00000000000000001e-262 or 5.0000000000000004e-19 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. distribute-rgt1-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in eps around 0 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   if 1.00000000000000001e-262 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5.0000000000000005e-193

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-define 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 99.2%

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

   7. Taylor expanded in x around inf 100.0%

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

   if 5e-108 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 5.0000000000000004e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-define 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 10^{-262}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-193}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-108}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ t_1 := 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ t_2 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ t_3 := {\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-270}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-55}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 12000000:\\ \;\;\;\;\varepsilon \cdot \sqrt[3]{125 \cdot {x}^{12}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow eps 4.0) (+ eps (* x 5.0))))
        (t_1 (* 5.0 (* eps (pow x 4.0))))
        (t_2 (* eps (* 5.0 (pow x 4.0))))
        (t_3 (* (pow eps 4.0) (* x (+ 5.0 (/ eps x))))))
   (if (<= x -4.8e-52)
     (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))
     (if (<= x -4e-78)
       t_0
       (if (<= x 1e-270)
         (pow eps 5.0)
         (if (<= x 1.5e-55)
           t_3
           (if (<= x 3e-45)
             (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
             (if (<= x 1.22e-26)
               t_2
               (if (<= x 1.45e-23)
                 t_3
                 (if (<= x 1.15e-18)
                   t_1
                   (if (<= x 1.16e-18)
                     t_0
                     (if (<= x 5e-7)
                       t_2
                       (if (<= x 12000000.0)
                         (* eps (cbrt (* 125.0 (pow x 12.0))))
                         t_1)))))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, 4.0) * (eps + (x * 5.0));
	double t_1 = 5.0 * (eps * pow(x, 4.0));
	double t_2 = eps * (5.0 * pow(x, 4.0));
	double t_3 = pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else if (x <= -4e-78) {
		tmp = t_0;
	} else if (x <= 1e-270) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1.5e-55) {
		tmp = t_3;
	} else if (x <= 3e-45) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (x <= 1.22e-26) {
		tmp = t_2;
	} else if (x <= 1.45e-23) {
		tmp = t_3;
	} else if (x <= 1.15e-18) {
		tmp = t_1;
	} else if (x <= 1.16e-18) {
		tmp = t_0;
	} else if (x <= 5e-7) {
		tmp = t_2;
	} else if (x <= 12000000.0) {
		tmp = eps * cbrt((125.0 * pow(x, 12.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	double t_1 = 5.0 * (eps * Math.pow(x, 4.0));
	double t_2 = eps * (5.0 * Math.pow(x, 4.0));
	double t_3 = Math.pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else if (x <= -4e-78) {
		tmp = t_0;
	} else if (x <= 1e-270) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1.5e-55) {
		tmp = t_3;
	} else if (x <= 3e-45) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (x <= 1.22e-26) {
		tmp = t_2;
	} else if (x <= 1.45e-23) {
		tmp = t_3;
	} else if (x <= 1.15e-18) {
		tmp = t_1;
	} else if (x <= 1.16e-18) {
		tmp = t_0;
	} else if (x <= 5e-7) {
		tmp = t_2;
	} else if (x <= 12000000.0) {
		tmp = eps * Math.cbrt((125.0 * Math.pow(x, 12.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)))
	t_1 = Float64(5.0 * Float64(eps * (x ^ 4.0)))
	t_2 = Float64(eps * Float64(5.0 * (x ^ 4.0)))
	t_3 = Float64((eps ^ 4.0) * Float64(x * Float64(5.0 + Float64(eps / x))))
	tmp = 0.0
	if (x <= -4.8e-52)
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	elseif (x <= -4e-78)
		tmp = t_0;
	elseif (x <= 1e-270)
		tmp = eps ^ 5.0;
	elseif (x <= 1.5e-55)
		tmp = t_3;
	elseif (x <= 3e-45)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	elseif (x <= 1.22e-26)
		tmp = t_2;
	elseif (x <= 1.45e-23)
		tmp = t_3;
	elseif (x <= 1.15e-18)
		tmp = t_1;
	elseif (x <= 1.16e-18)
		tmp = t_0;
	elseif (x <= 5e-7)
		tmp = t_2;
	elseif (x <= 12000000.0)
		tmp = Float64(eps * cbrt(Float64(125.0 * (x ^ 12.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-52], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-78], t$95$0, If[LessEqual[x, 1e-270], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1.5e-55], t$95$3, If[LessEqual[x, 3e-45], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-26], t$95$2, If[LessEqual[x, 1.45e-23], t$95$3, If[LessEqual[x, 1.15e-18], t$95$1, If[LessEqual[x, 1.16e-18], t$95$0, If[LessEqual[x, 5e-7], t$95$2, If[LessEqual[x, 12000000.0], N[(eps * N[Power[N[(125.0 * N[Power[x, 12.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if x < -4.8000000000000003e-52

   1. Initial program 46.2%

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

   3. Taylor expanded in x around -inf 95.1%

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

      1. +-commutative 95.1%

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

      2. associate-+r+ 95.1%

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

      3. mul-1-neg 95.1%

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

      4. unsub-neg 95.1%

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

      5. distribute-rgt1-in 95.1%

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

      6. metadata-eval 95.1%

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

      7. *-commutative 95.1%

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

   5. Simplified 95.1%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in eps around 0 95.1%

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

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   if -4.8000000000000003e-52 < x < -4e-78 or 1.15e-18 < x < 1.16e-18

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.2%

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

      1. fma-define 97.2%

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

      2. distribute-lft1-in 97.2%

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

      3. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in eps around 0 97.2%

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

   if -4e-78 < x < 1e-270

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1e-270 < x < 1.50000000000000008e-55 or 1.22e-26 < x < 1.4500000000000001e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. fma-define 99.2%

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

      2. distribute-lft1-in 99.2%

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

      3. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in eps around 0 99.2%

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

   7. Taylor expanded in x around inf 99.2%

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

   if 1.50000000000000008e-55 < x < 3.00000000000000011e-45

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if 3.00000000000000011e-45 < x < 1.22e-26 or 1.16e-18 < x < 4.99999999999999977e-7

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if 1.4500000000000001e-23 < x < 1.15e-18 or 1.2e7 < x

   1. Initial program 20.4%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in eps around 0 100.0%

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

   if 4.99999999999999977e-7 < x < 1.2e7

   1. Initial program 6.0%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-rgt1-in 99.2%

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

      3. metadata-eval 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 98.8%

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

   5. Simplified 98.8%

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

      1. add-sqr-sqrt 99.0%

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

      2. sqrt-unprod 98.8%

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

      3. *-commutative 98.8%

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

      4. *-commutative 98.8%

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

      5. swap-sqr 98.8%

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

      6. pow-prod-up 99.6%

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

      7. metadata-eval 99.6%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 99.6%

         \[\leadsto \varepsilon \cdot \color{blue}{\sqrt[3]{\left(\sqrt{{x}^{8} \cdot 25} \cdot \sqrt{{x}^{8} \cdot 25}\right) \cdot \sqrt{{x}^{8} \cdot 25}}} \]

      2. add-sqr-sqrt 99.2%

         \[\leadsto \varepsilon \cdot \sqrt[3]{\color{blue}{\left({x}^{8} \cdot 25\right)} \cdot \sqrt{{x}^{8} \cdot 25}} \]

      3. pow1 99.2%

         \[\leadsto \varepsilon \cdot \sqrt[3]{\color{blue}{{\left({x}^{8} \cdot 25\right)}^{1}} \cdot \sqrt{{x}^{8} \cdot 25}} \]

      4. pow1/2 99.2%

         \[\leadsto \varepsilon \cdot \sqrt[3]{{\left({x}^{8} \cdot 25\right)}^{1} \cdot \color{blue}{{\left({x}^{8} \cdot 25\right)}^{0.5}}} \]

      5. pow-prod-up 99.2%

         \[\leadsto \varepsilon \cdot \sqrt[3]{\color{blue}{{\left({x}^{8} \cdot 25\right)}^{\left(1 + 0.5\right)}}} \]

      6. metadata-eval 99.2%

         \[\leadsto \varepsilon \cdot \sqrt[3]{{\left({x}^{8} \cdot 25\right)}^{\color{blue}{1.5}}} \]

   9. Applied egg-rr 99.2%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt[3]{{\left({x}^{8} \cdot 25\right)}^{1.5}}} \]

   10. Taylor expanded in x around 0 100.0%

       \[\leadsto \varepsilon \cdot \sqrt[3]{\color{blue}{125 \cdot {x}^{12}}} \]
3. Recombined 8 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 10^{-270}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-55}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 12000000:\\ \;\;\;\;\varepsilon \cdot \sqrt[3]{125 \cdot {x}^{12}}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ t_1 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ t_2 := {\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{-51}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-268}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow eps 4.0) (+ eps (* x 5.0))))
        (t_1 (* eps (* 5.0 (pow x 4.0))))
        (t_2 (* (pow eps 4.0) (* x (+ 5.0 (/ eps x))))))
   (if (<= x -1.12e-51)
     (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))
     (if (<= x -4e-78)
       t_0
       (if (<= x 4e-268)
         (pow eps 5.0)
         (if (<= x 1e-55)
           t_2
           (if (<= x 2.15e-45)
             (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
             (if (<= x 1.22e-26)
               t_1
               (if (<= x 1.45e-23)
                 t_2
                 (if (<= x 1.15e-18)
                   (* 5.0 (* eps (pow x 4.0)))
                   (if (<= x 1.16e-18)
                     t_0
                     (if (<= x 8e-8)
                       t_1
                       (* (pow x 4.0) (* eps 5.0))))))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, 4.0) * (eps + (x * 5.0));
	double t_1 = eps * (5.0 * pow(x, 4.0));
	double t_2 = pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double tmp;
	if (x <= -1.12e-51) {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else if (x <= -4e-78) {
		tmp = t_0;
	} else if (x <= 4e-268) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1e-55) {
		tmp = t_2;
	} else if (x <= 2.15e-45) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (x <= 1.22e-26) {
		tmp = t_1;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_0;
	} else if (x <= 8e-8) {
		tmp = t_1;
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    t_1 = eps * (5.0d0 * (x ** 4.0d0))
    t_2 = (eps ** 4.0d0) * (x * (5.0d0 + (eps / x)))
    if (x <= (-1.12d-51)) then
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / x))))
    else if (x <= (-4d-78)) then
        tmp = t_0
    else if (x <= 4d-268) then
        tmp = eps ** 5.0d0
    else if (x <= 1d-55) then
        tmp = t_2
    else if (x <= 2.15d-45) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else if (x <= 1.22d-26) then
        tmp = t_1
    else if (x <= 1.45d-23) then
        tmp = t_2
    else if (x <= 1.15d-18) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.16d-18) then
        tmp = t_0
    else if (x <= 8d-8) then
        tmp = t_1
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	double t_1 = eps * (5.0 * Math.pow(x, 4.0));
	double t_2 = Math.pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double tmp;
	if (x <= -1.12e-51) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else if (x <= -4e-78) {
		tmp = t_0;
	} else if (x <= 4e-268) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1e-55) {
		tmp = t_2;
	} else if (x <= 2.15e-45) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (x <= 1.22e-26) {
		tmp = t_1;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_0;
	} else if (x <= 8e-8) {
		tmp = t_1;
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(eps, 4.0) * (eps + (x * 5.0))
	t_1 = eps * (5.0 * math.pow(x, 4.0))
	t_2 = math.pow(eps, 4.0) * (x * (5.0 + (eps / x)))
	tmp = 0
	if x <= -1.12e-51:
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	elif x <= -4e-78:
		tmp = t_0
	elif x <= 4e-268:
		tmp = math.pow(eps, 5.0)
	elif x <= 1e-55:
		tmp = t_2
	elif x <= 2.15e-45:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	elif x <= 1.22e-26:
		tmp = t_1
	elif x <= 1.45e-23:
		tmp = t_2
	elif x <= 1.15e-18:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.16e-18:
		tmp = t_0
	elif x <= 8e-8:
		tmp = t_1
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)))
	t_1 = Float64(eps * Float64(5.0 * (x ^ 4.0)))
	t_2 = Float64((eps ^ 4.0) * Float64(x * Float64(5.0 + Float64(eps / x))))
	tmp = 0.0
	if (x <= -1.12e-51)
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	elseif (x <= -4e-78)
		tmp = t_0;
	elseif (x <= 4e-268)
		tmp = eps ^ 5.0;
	elseif (x <= 1e-55)
		tmp = t_2;
	elseif (x <= 2.15e-45)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	elseif (x <= 1.22e-26)
		tmp = t_1;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.16e-18)
		tmp = t_0;
	elseif (x <= 8e-8)
		tmp = t_1;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps ^ 4.0) * (eps + (x * 5.0));
	t_1 = eps * (5.0 * (x ^ 4.0));
	t_2 = (eps ^ 4.0) * (x * (5.0 + (eps / x)));
	tmp = 0.0;
	if (x <= -1.12e-51)
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	elseif (x <= -4e-78)
		tmp = t_0;
	elseif (x <= 4e-268)
		tmp = eps ^ 5.0;
	elseif (x <= 1e-55)
		tmp = t_2;
	elseif (x <= 2.15e-45)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	elseif (x <= 1.22e-26)
		tmp = t_1;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.16e-18)
		tmp = t_0;
	elseif (x <= 8e-8)
		tmp = t_1;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.12e-51], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-78], t$95$0, If[LessEqual[x, 4e-268], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1e-55], t$95$2, If[LessEqual[x, 2.15e-45], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-26], t$95$1, If[LessEqual[x, 1.45e-23], t$95$2, If[LessEqual[x, 1.15e-18], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-18], t$95$0, If[LessEqual[x, 8e-8], t$95$1, N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if x < -1.11999999999999998e-51

   1. Initial program 46.2%

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

   3. Taylor expanded in x around -inf 95.1%

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

      1. +-commutative 95.1%

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

      2. associate-+r+ 95.1%

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

      3. mul-1-neg 95.1%

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

      4. unsub-neg 95.1%

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

      5. distribute-rgt1-in 95.1%

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

      6. metadata-eval 95.1%

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

      7. *-commutative 95.1%

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

   5. Simplified 95.1%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in eps around 0 95.1%

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

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   if -1.11999999999999998e-51 < x < -4e-78 or 1.15e-18 < x < 1.16e-18

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.2%

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

      1. fma-define 97.2%

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

      2. distribute-lft1-in 97.2%

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

      3. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in eps around 0 97.2%

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

   if -4e-78 < x < 3.99999999999999983e-268

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 3.99999999999999983e-268 < x < 9.99999999999999995e-56 or 1.22e-26 < x < 1.4500000000000001e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. fma-define 99.2%

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

      2. distribute-lft1-in 99.2%

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

      3. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in eps around 0 99.2%

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

   7. Taylor expanded in x around inf 99.2%

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

   if 9.99999999999999995e-56 < x < 2.1499999999999999e-45

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if 2.1499999999999999e-45 < x < 1.22e-26 or 1.16e-18 < x < 8.0000000000000002e-8

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if 1.4500000000000001e-23 < x < 1.15e-18

   1. Initial program 28.5%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

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

   if 8.0000000000000002e-8 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. distribute-rgt1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(5 \cdot \varepsilon\right)} \]
3. Recombined 8 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-51}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-268}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ t_1 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ t_2 := {\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ t_3 := {\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 10^{-270}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow x 4.0) (* eps 5.0)))
        (t_1 (* eps (* 5.0 (pow x 4.0))))
        (t_2 (* (pow eps 4.0) (* x (+ 5.0 (/ eps x)))))
        (t_3 (* (pow eps 4.0) (+ eps (* x 5.0)))))
   (if (<= x -1.5e-52)
     t_0
     (if (<= x -4e-78)
       t_3
       (if (<= x 1e-270)
         (pow eps 5.0)
         (if (<= x 1e-55)
           t_2
           (if (<= x 1.35e-45)
             (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
             (if (<= x 1.22e-26)
               t_1
               (if (<= x 1.45e-23)
                 t_2
                 (if (<= x 1.15e-18)
                   (* 5.0 (* eps (pow x 4.0)))
                   (if (<= x 1.16e-18) t_3 (if (<= x 3e-9) t_1 t_0))))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(x, 4.0) * (eps * 5.0);
	double t_1 = eps * (5.0 * pow(x, 4.0));
	double t_2 = pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double t_3 = pow(eps, 4.0) * (eps + (x * 5.0));
	double tmp;
	if (x <= -1.5e-52) {
		tmp = t_0;
	} else if (x <= -4e-78) {
		tmp = t_3;
	} else if (x <= 1e-270) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1e-55) {
		tmp = t_2;
	} else if (x <= 1.35e-45) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (x <= 1.22e-26) {
		tmp = t_1;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_3;
	} else if (x <= 3e-9) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (x ** 4.0d0) * (eps * 5.0d0)
    t_1 = eps * (5.0d0 * (x ** 4.0d0))
    t_2 = (eps ** 4.0d0) * (x * (5.0d0 + (eps / x)))
    t_3 = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    if (x <= (-1.5d-52)) then
        tmp = t_0
    else if (x <= (-4d-78)) then
        tmp = t_3
    else if (x <= 1d-270) then
        tmp = eps ** 5.0d0
    else if (x <= 1d-55) then
        tmp = t_2
    else if (x <= 1.35d-45) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else if (x <= 1.22d-26) then
        tmp = t_1
    else if (x <= 1.45d-23) then
        tmp = t_2
    else if (x <= 1.15d-18) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.16d-18) then
        tmp = t_3
    else if (x <= 3d-9) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(x, 4.0) * (eps * 5.0);
	double t_1 = eps * (5.0 * Math.pow(x, 4.0));
	double t_2 = Math.pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double t_3 = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	double tmp;
	if (x <= -1.5e-52) {
		tmp = t_0;
	} else if (x <= -4e-78) {
		tmp = t_3;
	} else if (x <= 1e-270) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1e-55) {
		tmp = t_2;
	} else if (x <= 1.35e-45) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (x <= 1.22e-26) {
		tmp = t_1;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_3;
	} else if (x <= 3e-9) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(x, 4.0) * (eps * 5.0)
	t_1 = eps * (5.0 * math.pow(x, 4.0))
	t_2 = math.pow(eps, 4.0) * (x * (5.0 + (eps / x)))
	t_3 = math.pow(eps, 4.0) * (eps + (x * 5.0))
	tmp = 0
	if x <= -1.5e-52:
		tmp = t_0
	elif x <= -4e-78:
		tmp = t_3
	elif x <= 1e-270:
		tmp = math.pow(eps, 5.0)
	elif x <= 1e-55:
		tmp = t_2
	elif x <= 1.35e-45:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	elif x <= 1.22e-26:
		tmp = t_1
	elif x <= 1.45e-23:
		tmp = t_2
	elif x <= 1.15e-18:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.16e-18:
		tmp = t_3
	elif x <= 3e-9:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((x ^ 4.0) * Float64(eps * 5.0))
	t_1 = Float64(eps * Float64(5.0 * (x ^ 4.0)))
	t_2 = Float64((eps ^ 4.0) * Float64(x * Float64(5.0 + Float64(eps / x))))
	t_3 = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)))
	tmp = 0.0
	if (x <= -1.5e-52)
		tmp = t_0;
	elseif (x <= -4e-78)
		tmp = t_3;
	elseif (x <= 1e-270)
		tmp = eps ^ 5.0;
	elseif (x <= 1e-55)
		tmp = t_2;
	elseif (x <= 1.35e-45)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	elseif (x <= 1.22e-26)
		tmp = t_1;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.16e-18)
		tmp = t_3;
	elseif (x <= 3e-9)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x ^ 4.0) * (eps * 5.0);
	t_1 = eps * (5.0 * (x ^ 4.0));
	t_2 = (eps ^ 4.0) * (x * (5.0 + (eps / x)));
	t_3 = (eps ^ 4.0) * (eps + (x * 5.0));
	tmp = 0.0;
	if (x <= -1.5e-52)
		tmp = t_0;
	elseif (x <= -4e-78)
		tmp = t_3;
	elseif (x <= 1e-270)
		tmp = eps ^ 5.0;
	elseif (x <= 1e-55)
		tmp = t_2;
	elseif (x <= 1.35e-45)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	elseif (x <= 1.22e-26)
		tmp = t_1;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.16e-18)
		tmp = t_3;
	elseif (x <= 3e-9)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e-52], t$95$0, If[LessEqual[x, -4e-78], t$95$3, If[LessEqual[x, 1e-270], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1e-55], t$95$2, If[LessEqual[x, 1.35e-45], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-26], t$95$1, If[LessEqual[x, 1.45e-23], t$95$2, If[LessEqual[x, 1.15e-18], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-18], t$95$3, If[LessEqual[x, 3e-9], t$95$1, t$95$0]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -1.5e-52 or 2.99999999999999998e-9 < x

   1. Initial program 37.1%

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

   3. Taylor expanded in x around inf 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. metadata-eval 94.8%

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

   5. Simplified 94.8%

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

   if -1.5e-52 < x < -4e-78 or 1.15e-18 < x < 1.16e-18

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.2%

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

      1. fma-define 97.2%

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

      2. distribute-lft1-in 97.2%

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

      3. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in eps around 0 97.2%

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

   if -4e-78 < x < 1e-270

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1e-270 < x < 9.99999999999999995e-56 or 1.22e-26 < x < 1.4500000000000001e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. fma-define 99.2%

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

      2. distribute-lft1-in 99.2%

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

      3. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in eps around 0 99.2%

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

   7. Taylor expanded in x around inf 99.2%

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

   if 9.99999999999999995e-56 < x < 1.34999999999999992e-45

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if 1.34999999999999992e-45 < x < 1.22e-26 or 1.16e-18 < x < 2.99999999999999998e-9

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if 1.4500000000000001e-23 < x < 1.15e-18

   1. Initial program 28.5%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-52}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 10^{-270}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ t_1 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ t_2 := {\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ t_3 := {\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-268}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (pow x 4.0))))
        (t_1 (* (pow x 4.0) (* eps 5.0)))
        (t_2 (* (pow eps 4.0) (* x (+ 5.0 (/ eps x)))))
        (t_3 (* (pow eps 4.0) (+ eps (* x 5.0)))))
   (if (<= x -1.6e-52)
     t_1
     (if (<= x -4e-78)
       t_3
       (if (<= x 4e-268)
         (pow eps 5.0)
         (if (<= x 1e-55)
           t_2
           (if (<= x 6.8e-45)
             t_3
             (if (<= x 1.22e-26)
               t_0
               (if (<= x 1.45e-23)
                 t_2
                 (if (<= x 1.15e-18)
                   (* 5.0 (* eps (pow x 4.0)))
                   (if (<= x 1.16e-18)
                     t_3
                     (if (<= x 3.05e-9) t_0 t_1))))))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * pow(x, 4.0));
	double t_1 = pow(x, 4.0) * (eps * 5.0);
	double t_2 = pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double t_3 = pow(eps, 4.0) * (eps + (x * 5.0));
	double tmp;
	if (x <= -1.6e-52) {
		tmp = t_1;
	} else if (x <= -4e-78) {
		tmp = t_3;
	} else if (x <= 4e-268) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1e-55) {
		tmp = t_2;
	} else if (x <= 6.8e-45) {
		tmp = t_3;
	} else if (x <= 1.22e-26) {
		tmp = t_0;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_3;
	} else if (x <= 3.05e-9) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = eps * (5.0d0 * (x ** 4.0d0))
    t_1 = (x ** 4.0d0) * (eps * 5.0d0)
    t_2 = (eps ** 4.0d0) * (x * (5.0d0 + (eps / x)))
    t_3 = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    if (x <= (-1.6d-52)) then
        tmp = t_1
    else if (x <= (-4d-78)) then
        tmp = t_3
    else if (x <= 4d-268) then
        tmp = eps ** 5.0d0
    else if (x <= 1d-55) then
        tmp = t_2
    else if (x <= 6.8d-45) then
        tmp = t_3
    else if (x <= 1.22d-26) then
        tmp = t_0
    else if (x <= 1.45d-23) then
        tmp = t_2
    else if (x <= 1.15d-18) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.16d-18) then
        tmp = t_3
    else if (x <= 3.05d-9) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * Math.pow(x, 4.0));
	double t_1 = Math.pow(x, 4.0) * (eps * 5.0);
	double t_2 = Math.pow(eps, 4.0) * (x * (5.0 + (eps / x)));
	double t_3 = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	double tmp;
	if (x <= -1.6e-52) {
		tmp = t_1;
	} else if (x <= -4e-78) {
		tmp = t_3;
	} else if (x <= 4e-268) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1e-55) {
		tmp = t_2;
	} else if (x <= 6.8e-45) {
		tmp = t_3;
	} else if (x <= 1.22e-26) {
		tmp = t_0;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_3;
	} else if (x <= 3.05e-9) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * math.pow(x, 4.0))
	t_1 = math.pow(x, 4.0) * (eps * 5.0)
	t_2 = math.pow(eps, 4.0) * (x * (5.0 + (eps / x)))
	t_3 = math.pow(eps, 4.0) * (eps + (x * 5.0))
	tmp = 0
	if x <= -1.6e-52:
		tmp = t_1
	elif x <= -4e-78:
		tmp = t_3
	elif x <= 4e-268:
		tmp = math.pow(eps, 5.0)
	elif x <= 1e-55:
		tmp = t_2
	elif x <= 6.8e-45:
		tmp = t_3
	elif x <= 1.22e-26:
		tmp = t_0
	elif x <= 1.45e-23:
		tmp = t_2
	elif x <= 1.15e-18:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.16e-18:
		tmp = t_3
	elif x <= 3.05e-9:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * (x ^ 4.0)))
	t_1 = Float64((x ^ 4.0) * Float64(eps * 5.0))
	t_2 = Float64((eps ^ 4.0) * Float64(x * Float64(5.0 + Float64(eps / x))))
	t_3 = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)))
	tmp = 0.0
	if (x <= -1.6e-52)
		tmp = t_1;
	elseif (x <= -4e-78)
		tmp = t_3;
	elseif (x <= 4e-268)
		tmp = eps ^ 5.0;
	elseif (x <= 1e-55)
		tmp = t_2;
	elseif (x <= 6.8e-45)
		tmp = t_3;
	elseif (x <= 1.22e-26)
		tmp = t_0;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.16e-18)
		tmp = t_3;
	elseif (x <= 3.05e-9)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (5.0 * (x ^ 4.0));
	t_1 = (x ^ 4.0) * (eps * 5.0);
	t_2 = (eps ^ 4.0) * (x * (5.0 + (eps / x)));
	t_3 = (eps ^ 4.0) * (eps + (x * 5.0));
	tmp = 0.0;
	if (x <= -1.6e-52)
		tmp = t_1;
	elseif (x <= -4e-78)
		tmp = t_3;
	elseif (x <= 4e-268)
		tmp = eps ^ 5.0;
	elseif (x <= 1e-55)
		tmp = t_2;
	elseif (x <= 6.8e-45)
		tmp = t_3;
	elseif (x <= 1.22e-26)
		tmp = t_0;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.16e-18)
		tmp = t_3;
	elseif (x <= 3.05e-9)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e-52], t$95$1, If[LessEqual[x, -4e-78], t$95$3, If[LessEqual[x, 4e-268], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1e-55], t$95$2, If[LessEqual[x, 6.8e-45], t$95$3, If[LessEqual[x, 1.22e-26], t$95$0, If[LessEqual[x, 1.45e-23], t$95$2, If[LessEqual[x, 1.15e-18], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-18], t$95$3, If[LessEqual[x, 3.05e-9], t$95$0, t$95$1]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.60000000000000005e-52 or 3.05e-9 < x

   1. Initial program 37.1%

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

   3. Taylor expanded in x around inf 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. metadata-eval 94.8%

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

   5. Simplified 94.8%

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

   if -1.60000000000000005e-52 < x < -4e-78 or 9.99999999999999995e-56 < x < 6.80000000000000008e-45 or 1.15e-18 < x < 1.16e-18

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.8%

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

      1. fma-define 97.8%

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

      2. distribute-lft1-in 97.8%

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

      3. metadata-eval 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in eps around 0 97.8%

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

   if -4e-78 < x < 3.99999999999999983e-268

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 3.99999999999999983e-268 < x < 9.99999999999999995e-56 or 1.22e-26 < x < 1.4500000000000001e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. fma-define 99.2%

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

      2. distribute-lft1-in 99.2%

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

      3. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in eps around 0 99.2%

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

   7. Taylor expanded in x around inf 99.2%

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

   if 6.80000000000000008e-45 < x < 1.22e-26 or 1.16e-18 < x < 3.05e-9

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if 1.4500000000000001e-23 < x < 1.15e-18

   1. Initial program 28.5%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-268}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ t_1 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ t_2 := {\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 10^{-270}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow x 4.0) (* eps 5.0)))
        (t_1 (* eps (* 5.0 (pow x 4.0))))
        (t_2 (* (pow eps 4.0) (+ eps (* x 5.0)))))
   (if (<= x -1.2e-51)
     t_0
     (if (<= x -4e-78)
       t_2
       (if (<= x 1e-270)
         (pow eps 5.0)
         (if (<= x 1.65e-45)
           t_2
           (if (<= x 1.22e-26)
             t_1
             (if (<= x 1.45e-23)
               t_2
               (if (<= x 1.15e-18)
                 (* 5.0 (* eps (pow x 4.0)))
                 (if (<= x 1.16e-18) t_2 (if (<= x 1.05e-7) t_1 t_0)))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(x, 4.0) * (eps * 5.0);
	double t_1 = eps * (5.0 * pow(x, 4.0));
	double t_2 = pow(eps, 4.0) * (eps + (x * 5.0));
	double tmp;
	if (x <= -1.2e-51) {
		tmp = t_0;
	} else if (x <= -4e-78) {
		tmp = t_2;
	} else if (x <= 1e-270) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1.65e-45) {
		tmp = t_2;
	} else if (x <= 1.22e-26) {
		tmp = t_1;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_2;
	} else if (x <= 1.05e-7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x ** 4.0d0) * (eps * 5.0d0)
    t_1 = eps * (5.0d0 * (x ** 4.0d0))
    t_2 = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    if (x <= (-1.2d-51)) then
        tmp = t_0
    else if (x <= (-4d-78)) then
        tmp = t_2
    else if (x <= 1d-270) then
        tmp = eps ** 5.0d0
    else if (x <= 1.65d-45) then
        tmp = t_2
    else if (x <= 1.22d-26) then
        tmp = t_1
    else if (x <= 1.45d-23) then
        tmp = t_2
    else if (x <= 1.15d-18) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.16d-18) then
        tmp = t_2
    else if (x <= 1.05d-7) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(x, 4.0) * (eps * 5.0);
	double t_1 = eps * (5.0 * Math.pow(x, 4.0));
	double t_2 = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	double tmp;
	if (x <= -1.2e-51) {
		tmp = t_0;
	} else if (x <= -4e-78) {
		tmp = t_2;
	} else if (x <= 1e-270) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1.65e-45) {
		tmp = t_2;
	} else if (x <= 1.22e-26) {
		tmp = t_1;
	} else if (x <= 1.45e-23) {
		tmp = t_2;
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = t_2;
	} else if (x <= 1.05e-7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(x, 4.0) * (eps * 5.0)
	t_1 = eps * (5.0 * math.pow(x, 4.0))
	t_2 = math.pow(eps, 4.0) * (eps + (x * 5.0))
	tmp = 0
	if x <= -1.2e-51:
		tmp = t_0
	elif x <= -4e-78:
		tmp = t_2
	elif x <= 1e-270:
		tmp = math.pow(eps, 5.0)
	elif x <= 1.65e-45:
		tmp = t_2
	elif x <= 1.22e-26:
		tmp = t_1
	elif x <= 1.45e-23:
		tmp = t_2
	elif x <= 1.15e-18:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.16e-18:
		tmp = t_2
	elif x <= 1.05e-7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((x ^ 4.0) * Float64(eps * 5.0))
	t_1 = Float64(eps * Float64(5.0 * (x ^ 4.0)))
	t_2 = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)))
	tmp = 0.0
	if (x <= -1.2e-51)
		tmp = t_0;
	elseif (x <= -4e-78)
		tmp = t_2;
	elseif (x <= 1e-270)
		tmp = eps ^ 5.0;
	elseif (x <= 1.65e-45)
		tmp = t_2;
	elseif (x <= 1.22e-26)
		tmp = t_1;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.16e-18)
		tmp = t_2;
	elseif (x <= 1.05e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x ^ 4.0) * (eps * 5.0);
	t_1 = eps * (5.0 * (x ^ 4.0));
	t_2 = (eps ^ 4.0) * (eps + (x * 5.0));
	tmp = 0.0;
	if (x <= -1.2e-51)
		tmp = t_0;
	elseif (x <= -4e-78)
		tmp = t_2;
	elseif (x <= 1e-270)
		tmp = eps ^ 5.0;
	elseif (x <= 1.65e-45)
		tmp = t_2;
	elseif (x <= 1.22e-26)
		tmp = t_1;
	elseif (x <= 1.45e-23)
		tmp = t_2;
	elseif (x <= 1.15e-18)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.16e-18)
		tmp = t_2;
	elseif (x <= 1.05e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e-51], t$95$0, If[LessEqual[x, -4e-78], t$95$2, If[LessEqual[x, 1e-270], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1.65e-45], t$95$2, If[LessEqual[x, 1.22e-26], t$95$1, If[LessEqual[x, 1.45e-23], t$95$2, If[LessEqual[x, 1.15e-18], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-18], t$95$2, If[LessEqual[x, 1.05e-7], t$95$1, t$95$0]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.2e-51 or 1.05e-7 < x

   1. Initial program 37.1%

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

   3. Taylor expanded in x around inf 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. metadata-eval 94.8%

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

   5. Simplified 94.8%

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

   if -1.2e-51 < x < -4e-78 or 1e-270 < x < 1.65e-45 or 1.22e-26 < x < 1.4500000000000001e-23 or 1.15e-18 < x < 1.16e-18

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.0%

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

      1. fma-define 99.0%

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

      2. distribute-lft1-in 99.0%

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

      3. metadata-eval 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in eps around 0 99.0%

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

   if -4e-78 < x < 1e-270

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1.65e-45 < x < 1.22e-26 or 1.16e-18 < x < 1.05e-7

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if 1.4500000000000001e-23 < x < 1.15e-18

   1. Initial program 28.5%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 10^{-270}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ t_1 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (pow x 4.0)))) (t_1 (* (pow x 4.0) (* eps 5.0))))
   (if (<= x -9e-53)
     t_1
     (if (<= x 6.8e-45)
       (pow eps 5.0)
       (if (<= x 1.22e-26)
         t_0
         (if (<= x 1.45e-23)
           (pow eps 5.0)
           (if (<= x 1.15e-18)
             (* 5.0 (* eps (pow x 4.0)))
             (if (<= x 1.16e-18)
               (pow eps 5.0)
               (if (<= x 5e-10) t_0 t_1)))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * pow(x, 4.0));
	double t_1 = pow(x, 4.0) * (eps * 5.0);
	double tmp;
	if (x <= -9e-53) {
		tmp = t_1;
	} else if (x <= 6.8e-45) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1.22e-26) {
		tmp = t_0;
	} else if (x <= 1.45e-23) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = pow(eps, 5.0);
	} else if (x <= 5e-10) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = eps * (5.0d0 * (x ** 4.0d0))
    t_1 = (x ** 4.0d0) * (eps * 5.0d0)
    if (x <= (-9d-53)) then
        tmp = t_1
    else if (x <= 6.8d-45) then
        tmp = eps ** 5.0d0
    else if (x <= 1.22d-26) then
        tmp = t_0
    else if (x <= 1.45d-23) then
        tmp = eps ** 5.0d0
    else if (x <= 1.15d-18) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.16d-18) then
        tmp = eps ** 5.0d0
    else if (x <= 5d-10) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * Math.pow(x, 4.0));
	double t_1 = Math.pow(x, 4.0) * (eps * 5.0);
	double tmp;
	if (x <= -9e-53) {
		tmp = t_1;
	} else if (x <= 6.8e-45) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1.22e-26) {
		tmp = t_0;
	} else if (x <= 1.45e-23) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1.15e-18) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.16e-18) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 5e-10) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * math.pow(x, 4.0))
	t_1 = math.pow(x, 4.0) * (eps * 5.0)
	tmp = 0
	if x <= -9e-53:
		tmp = t_1
	elif x <= 6.8e-45:
		tmp = math.pow(eps, 5.0)
	elif x <= 1.22e-26:
		tmp = t_0
	elif x <= 1.45e-23:
		tmp = math.pow(eps, 5.0)
	elif x <= 1.15e-18:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.16e-18:
		tmp = math.pow(eps, 5.0)
	elif x <= 5e-10:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * (x ^ 4.0)))
	t_1 = Float64((x ^ 4.0) * Float64(eps * 5.0))
	tmp = 0.0
	if (x <= -9e-53)
		tmp = t_1;
	elseif (x <= 6.8e-45)
		tmp = eps ^ 5.0;
	elseif (x <= 1.22e-26)
		tmp = t_0;
	elseif (x <= 1.45e-23)
		tmp = eps ^ 5.0;
	elseif (x <= 1.15e-18)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.16e-18)
		tmp = eps ^ 5.0;
	elseif (x <= 5e-10)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (5.0 * (x ^ 4.0));
	t_1 = (x ^ 4.0) * (eps * 5.0);
	tmp = 0.0;
	if (x <= -9e-53)
		tmp = t_1;
	elseif (x <= 6.8e-45)
		tmp = eps ^ 5.0;
	elseif (x <= 1.22e-26)
		tmp = t_0;
	elseif (x <= 1.45e-23)
		tmp = eps ^ 5.0;
	elseif (x <= 1.15e-18)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.16e-18)
		tmp = eps ^ 5.0;
	elseif (x <= 5e-10)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-53], t$95$1, If[LessEqual[x, 6.8e-45], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1.22e-26], t$95$0, If[LessEqual[x, 1.45e-23], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1.15e-18], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-18], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 5e-10], t$95$0, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.9999999999999997e-53 or 5.00000000000000031e-10 < x

   1. Initial program 37.1%

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

   3. Taylor expanded in x around inf 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. metadata-eval 94.8%

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

   5. Simplified 94.8%

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

   if -8.9999999999999997e-53 < x < 6.80000000000000008e-45 or 1.22e-26 < x < 1.4500000000000001e-23 or 1.15e-18 < x < 1.16e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 6.80000000000000008e-45 < x < 1.22e-26 or 1.16e-18 < x < 5.00000000000000031e-10

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if 1.4500000000000001e-23 < x < 1.15e-18

   1. Initial program 28.5%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-rgt1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-53}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ t_1 := 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-45}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;x \leq 14000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (pow x 4.0)))) (t_1 (* 5.0 (* eps (pow x 4.0)))))
   (if (<= x -6.8e-53)
     t_0
     (if (<= x 1.4e-45)
       (pow eps 5.0)
       (if (<= x 1.22e-26)
         t_0
         (if (<= x 1.45e-23)
           (pow eps 5.0)
           (if (<= x 1.15e-18)
             t_1
             (if (<= x 1.16e-18)
               (pow eps 5.0)
               (if (<= x 14000000.0) t_0 t_1)))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * pow(x, 4.0));
	double t_1 = 5.0 * (eps * pow(x, 4.0));
	double tmp;
	if (x <= -6.8e-53) {
		tmp = t_0;
	} else if (x <= 1.4e-45) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1.22e-26) {
		tmp = t_0;
	} else if (x <= 1.45e-23) {
		tmp = pow(eps, 5.0);
	} else if (x <= 1.15e-18) {
		tmp = t_1;
	} else if (x <= 1.16e-18) {
		tmp = pow(eps, 5.0);
	} else if (x <= 14000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = eps * (5.0d0 * (x ** 4.0d0))
    t_1 = 5.0d0 * (eps * (x ** 4.0d0))
    if (x <= (-6.8d-53)) then
        tmp = t_0
    else if (x <= 1.4d-45) then
        tmp = eps ** 5.0d0
    else if (x <= 1.22d-26) then
        tmp = t_0
    else if (x <= 1.45d-23) then
        tmp = eps ** 5.0d0
    else if (x <= 1.15d-18) then
        tmp = t_1
    else if (x <= 1.16d-18) then
        tmp = eps ** 5.0d0
    else if (x <= 14000000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * Math.pow(x, 4.0));
	double t_1 = 5.0 * (eps * Math.pow(x, 4.0));
	double tmp;
	if (x <= -6.8e-53) {
		tmp = t_0;
	} else if (x <= 1.4e-45) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1.22e-26) {
		tmp = t_0;
	} else if (x <= 1.45e-23) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 1.15e-18) {
		tmp = t_1;
	} else if (x <= 1.16e-18) {
		tmp = Math.pow(eps, 5.0);
	} else if (x <= 14000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * math.pow(x, 4.0))
	t_1 = 5.0 * (eps * math.pow(x, 4.0))
	tmp = 0
	if x <= -6.8e-53:
		tmp = t_0
	elif x <= 1.4e-45:
		tmp = math.pow(eps, 5.0)
	elif x <= 1.22e-26:
		tmp = t_0
	elif x <= 1.45e-23:
		tmp = math.pow(eps, 5.0)
	elif x <= 1.15e-18:
		tmp = t_1
	elif x <= 1.16e-18:
		tmp = math.pow(eps, 5.0)
	elif x <= 14000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * (x ^ 4.0)))
	t_1 = Float64(5.0 * Float64(eps * (x ^ 4.0)))
	tmp = 0.0
	if (x <= -6.8e-53)
		tmp = t_0;
	elseif (x <= 1.4e-45)
		tmp = eps ^ 5.0;
	elseif (x <= 1.22e-26)
		tmp = t_0;
	elseif (x <= 1.45e-23)
		tmp = eps ^ 5.0;
	elseif (x <= 1.15e-18)
		tmp = t_1;
	elseif (x <= 1.16e-18)
		tmp = eps ^ 5.0;
	elseif (x <= 14000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (5.0 * (x ^ 4.0));
	t_1 = 5.0 * (eps * (x ^ 4.0));
	tmp = 0.0;
	if (x <= -6.8e-53)
		tmp = t_0;
	elseif (x <= 1.4e-45)
		tmp = eps ^ 5.0;
	elseif (x <= 1.22e-26)
		tmp = t_0;
	elseif (x <= 1.45e-23)
		tmp = eps ^ 5.0;
	elseif (x <= 1.15e-18)
		tmp = t_1;
	elseif (x <= 1.16e-18)
		tmp = eps ^ 5.0;
	elseif (x <= 14000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e-53], t$95$0, If[LessEqual[x, 1.4e-45], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1.22e-26], t$95$0, If[LessEqual[x, 1.45e-23], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 1.15e-18], t$95$1, If[LessEqual[x, 1.16e-18], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[x, 14000000.0], t$95$0, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e-53 or 1.4000000000000001e-45 < x < 1.22e-26 or 1.16e-18 < x < 1.4e7

   1. Initial program 38.9%

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

   3. Taylor expanded in x around inf 95.5%

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

      1. *-commutative 95.5%

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

      2. distribute-rgt1-in 95.5%

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

      3. metadata-eval 95.5%

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

      4. *-commutative 95.5%

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

      5. associate-*r* 95.5%

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

   5. Simplified 95.5%

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

   if -6.8e-53 < x < 1.4000000000000001e-45 or 1.22e-26 < x < 1.4500000000000001e-23 or 1.15e-18 < x < 1.16e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 1.4500000000000001e-23 < x < 1.15e-18 or 1.4e7 < x

   1. Initial program 20.4%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in eps around 0 100.0%

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

Alternative 11: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-52} \lor \neg \left(x \leq 1.32 \cdot 10^{-45}\right) \land \left(x \leq 1.22 \cdot 10^{-26} \lor \neg \left(x \leq 1.45 \cdot 10^{-23}\right) \land \left(x \leq 1.15 \cdot 10^{-18} \lor \neg \left(x \leq 1.16 \cdot 10^{-18}\right)\right)\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.7e-52)
         (and (not (<= x 1.32e-45))
              (or (<= x 1.22e-26)
                  (and (not (<= x 1.45e-23))
                       (or (<= x 1.15e-18) (not (<= x 1.16e-18)))))))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.7e-52) || (!(x <= 1.32e-45) && ((x <= 1.22e-26) || (!(x <= 1.45e-23) && ((x <= 1.15e-18) || !(x <= 1.16e-18)))))) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5.7d-52)) .or. (.not. (x <= 1.32d-45)) .and. (x <= 1.22d-26) .or. (.not. (x <= 1.45d-23)) .and. (x <= 1.15d-18) .or. (.not. (x <= 1.16d-18))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5.7e-52) || (!(x <= 1.32e-45) && ((x <= 1.22e-26) || (!(x <= 1.45e-23) && ((x <= 1.15e-18) || !(x <= 1.16e-18)))))) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5.7e-52) or (not (x <= 1.32e-45) and ((x <= 1.22e-26) or (not (x <= 1.45e-23) and ((x <= 1.15e-18) or not (x <= 1.16e-18))))):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5.7e-52) || (!(x <= 1.32e-45) && ((x <= 1.22e-26) || (!(x <= 1.45e-23) && ((x <= 1.15e-18) || !(x <= 1.16e-18))))))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5.7e-52) || (~((x <= 1.32e-45)) && ((x <= 1.22e-26) || (~((x <= 1.45e-23)) && ((x <= 1.15e-18) || ~((x <= 1.16e-18)))))))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5.7e-52], And[N[Not[LessEqual[x, 1.32e-45]], $MachinePrecision], Or[LessEqual[x, 1.22e-26], And[N[Not[LessEqual[x, 1.45e-23]], $MachinePrecision], Or[LessEqual[x, 1.15e-18], N[Not[LessEqual[x, 1.16e-18]], $MachinePrecision]]]]]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6999999999999997e-52 or 1.32000000000000005e-45 < x < 1.22e-26 or 1.4500000000000001e-23 < x < 1.15e-18 or 1.16e-18 < x

   1. Initial program 36.0%

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

   3. Taylor expanded in x around inf 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-rgt1-in 96.2%

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

      3. metadata-eval 96.2%

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

      4. *-commutative 96.2%

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

      5. associate-*r* 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in eps around 0 96.0%

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

   if -5.6999999999999997e-52 < x < 1.32000000000000005e-45 or 1.22e-26 < x < 1.4500000000000001e-23 or 1.15e-18 < x < 1.16e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-52} \lor \neg \left(x \leq 1.32 \cdot 10^{-45}\right) \land \left(x \leq 1.22 \cdot 10^{-26} \lor \neg \left(x \leq 1.45 \cdot 10^{-23}\right) \land \left(x \leq 1.15 \cdot 10^{-18} \lor \neg \left(x \leq 1.16 \cdot 10^{-18}\right)\right)\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return eps ^ 5.0
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Power[eps, 5.0], $MachinePrecision]

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in x around 0 89.1%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))