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

Percentage Accurate: 88.3% → 97.7%

Time: 8.3s

Alternatives: 11

Speedup: 2.0×

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: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(\sqrt{25 \cdot {x}^{8}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right) + {\varepsilon}^{3}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-38)
   (* eps (+ (sqrt (* 25.0 (pow x 8.0))) (* eps (* (pow x 3.0) 10.0))))
   (if (<= x 4.7e-60)
     (+ (pow eps 5.0) (* x (* 5.0 (pow eps 4.0))))
     (fma
      (* eps 5.0)
      (pow x 4.0)
      (* x (* x (* 10.0 (+ (* x (* eps eps)) (pow eps 3.0)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-38) {
		tmp = eps * (sqrt((25.0 * pow(x, 8.0))) + (eps * (pow(x, 3.0) * 10.0)));
	} else if (x <= 4.7e-60) {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	} else {
		tmp = fma((eps * 5.0), pow(x, 4.0), (x * (x * (10.0 * ((x * (eps * eps)) + pow(eps, 3.0))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-38)
		tmp = Float64(eps * Float64(sqrt(Float64(25.0 * (x ^ 8.0))) + Float64(eps * Float64((x ^ 3.0) * 10.0))));
	elseif (x <= 4.7e-60)
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	else
		tmp = fma(Float64(eps * 5.0), (x ^ 4.0), Float64(x * Float64(x * Float64(10.0 * Float64(Float64(x * Float64(eps * eps)) + (eps ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-38], N[(eps * N[(N[Sqrt[N[(25.0 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.7e-60], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(x * N[(x * N[(10.0 * N[(N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.19999999999999966e-38

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. associate-*l* 99.7%

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

      4. distribute-lft-out 99.7%

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

      5. distribute-lft1-in 99.7%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(4 + 1\right) \cdot {x}^{4}} + \varepsilon \cdot \left(4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x\right)\right) \]

      6. metadata-eval 99.7%

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

      7. *-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. associate-*r* 99.7%

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

   4. Simplified 99.7%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.7%

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

      3. swap-sqr 99.5%

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

      4. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(\sqrt{\color{blue}{25} \cdot \left({x}^{4} \cdot {x}^{4}\right)} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

      5. pow-prod-up 100.0%

         \[\leadsto \varepsilon \cdot \left(\sqrt{25 \cdot \color{blue}{{x}^{\left(4 + 4\right)}}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

      6. metadata-eval 100.0%

         \[\leadsto \varepsilon \cdot \left(\sqrt{25 \cdot {x}^{\color{blue}{8}}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\sqrt{25 \cdot {x}^{8}}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

   if -6.19999999999999966e-38 < x < 4.7e-60

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   4. Simplified 99.5%

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

   if 4.7e-60 < x

   1. Initial program 39.2%

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

   2. Taylor expanded in x around inf 94.6%

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

      1. fma-def 94.7%

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

      2. distribute-lft1-in 94.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 94.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 94.7%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 94.7%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 94.7%

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

   5. Taylor expanded in x around 0 94.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10 + \color{blue}{\left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10}\right) \]

      4. unpow2 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      5. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(x \cdot x\right) \cdot {\varepsilon}^{3}\right)} \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      6. associate-*r* 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10\right)} + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      7. unpow2 94.7%

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

      8. *-commutative 94.7%

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

      9. associate-*r* 94.7%

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

      10. *-commutative 94.7%

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

      11. cube-mult 94.7%

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

      12. associate-*l* 94.7%

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

      13. *-commutative 94.7%

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

      14. distribute-lft-in 94.7%

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

      15. fma-udef 94.7%

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

      16. associate-*l* 94.7%

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

   7. Simplified 94.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(\sqrt{25 \cdot {x}^{8}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right) + {\varepsilon}^{3}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(\sqrt{25 \cdot {x}^{8}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-38)
   (* eps (+ (sqrt (* 25.0 (pow x 8.0))) (* eps (* (pow x 3.0) 10.0))))
   (if (<= x 4.7e-60)
     (+ (pow eps 5.0) (* x (* 5.0 (pow eps 4.0))))
     (+
      (* (* eps 5.0) (pow x 4.0))
      (* x (* 10.0 (* x (* (* eps eps) (+ x eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-38) {
		tmp = eps * (sqrt((25.0 * pow(x, 8.0))) + (eps * (pow(x, 3.0) * 10.0)));
	} else if (x <= 4.7e-60) {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	} else {
		tmp = ((eps * 5.0) * pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.2d-38)) then
        tmp = eps * (sqrt((25.0d0 * (x ** 8.0d0))) + (eps * ((x ** 3.0d0) * 10.0d0)))
    else if (x <= 4.7d-60) then
        tmp = (eps ** 5.0d0) + (x * (5.0d0 * (eps ** 4.0d0)))
    else
        tmp = ((eps * 5.0d0) * (x ** 4.0d0)) + (x * (10.0d0 * (x * ((eps * eps) * (x + eps)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-38) {
		tmp = eps * (Math.sqrt((25.0 * Math.pow(x, 8.0))) + (eps * (Math.pow(x, 3.0) * 10.0)));
	} else if (x <= 4.7e-60) {
		tmp = Math.pow(eps, 5.0) + (x * (5.0 * Math.pow(eps, 4.0)));
	} else {
		tmp = ((eps * 5.0) * Math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e-38:
		tmp = eps * (math.sqrt((25.0 * math.pow(x, 8.0))) + (eps * (math.pow(x, 3.0) * 10.0)))
	elif x <= 4.7e-60:
		tmp = math.pow(eps, 5.0) + (x * (5.0 * math.pow(eps, 4.0)))
	else:
		tmp = ((eps * 5.0) * math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-38)
		tmp = Float64(eps * Float64(sqrt(Float64(25.0 * (x ^ 8.0))) + Float64(eps * Float64((x ^ 3.0) * 10.0))));
	elseif (x <= 4.7e-60)
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(eps * 5.0) * (x ^ 4.0)) + Float64(x * Float64(10.0 * Float64(x * Float64(Float64(eps * eps) * Float64(x + eps))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e-38)
		tmp = eps * (sqrt((25.0 * (x ^ 8.0))) + (eps * ((x ^ 3.0) * 10.0)));
	elseif (x <= 4.7e-60)
		tmp = (eps ^ 5.0) + (x * (5.0 * (eps ^ 4.0)));
	else
		tmp = ((eps * 5.0) * (x ^ 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-38], N[(eps * N[(N[Sqrt[N[(25.0 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.7e-60], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000022e-38

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. associate-*l* 99.7%

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

      4. distribute-lft-out 99.7%

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

      5. distribute-lft1-in 99.7%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(4 + 1\right) \cdot {x}^{4}} + \varepsilon \cdot \left(4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x\right)\right) \]

      6. metadata-eval 99.7%

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

      7. *-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. associate-*r* 99.7%

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

   4. Simplified 99.7%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.7%

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

      3. swap-sqr 99.5%

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

      4. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(\sqrt{\color{blue}{25} \cdot \left({x}^{4} \cdot {x}^{4}\right)} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

      5. pow-prod-up 100.0%

         \[\leadsto \varepsilon \cdot \left(\sqrt{25 \cdot \color{blue}{{x}^{\left(4 + 4\right)}}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

      6. metadata-eval 100.0%

         \[\leadsto \varepsilon \cdot \left(\sqrt{25 \cdot {x}^{\color{blue}{8}}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\sqrt{25 \cdot {x}^{8}}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right) \]

   if -5.20000000000000022e-38 < x < 4.7e-60

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   4. Simplified 99.5%

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

   if 4.7e-60 < x

   1. Initial program 39.2%

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

   2. Taylor expanded in x around inf 94.6%

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

      1. fma-def 94.7%

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

      2. distribute-lft1-in 94.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 94.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 94.7%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 94.7%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 94.7%

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

   5. Taylor expanded in x around 0 94.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10 + \color{blue}{\left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10}\right) \]

      4. unpow2 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      5. *-commutative 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(x \cdot x\right) \cdot {\varepsilon}^{3}\right)} \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      6. associate-*r* 94.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10\right)} + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      7. unpow2 94.7%

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

      8. *-commutative 94.7%

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

      9. associate-*r* 94.7%

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

      10. *-commutative 94.7%

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

      11. cube-mult 94.7%

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

      12. associate-*l* 94.7%

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

      13. *-commutative 94.7%

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

      14. distribute-lft-in 94.7%

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

      15. fma-udef 94.7%

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

      16. associate-*l* 94.7%

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

   7. Simplified 94.7%

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

      1. fma-udef 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*l* 94.6%

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

      4. cube-mult 94.6%

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

      5. distribute-rgt-out 94.6%

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(\sqrt{25 \cdot {x}^{8}} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.2e-38) (not (<= x 4.7e-60)))
   (+
    (* (* eps 5.0) (pow x 4.0))
    (* x (* 10.0 (* x (* (* eps eps) (+ x eps))))))
   (+ (pow eps 5.0) (* x (* 5.0 (pow eps 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e-38) || !(x <= 4.7e-60)) {
		tmp = ((eps * 5.0) * pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	} else {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5.2d-38)) .or. (.not. (x <= 4.7d-60))) then
        tmp = ((eps * 5.0d0) * (x ** 4.0d0)) + (x * (10.0d0 * (x * ((eps * eps) * (x + eps)))))
    else
        tmp = (eps ** 5.0d0) + (x * (5.0d0 * (eps ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e-38) || !(x <= 4.7e-60)) {
		tmp = ((eps * 5.0) * Math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	} else {
		tmp = Math.pow(eps, 5.0) + (x * (5.0 * Math.pow(eps, 4.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5.2e-38) or not (x <= 4.7e-60):
		tmp = ((eps * 5.0) * math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))))
	else:
		tmp = math.pow(eps, 5.0) + (x * (5.0 * math.pow(eps, 4.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5.2e-38) || !(x <= 4.7e-60))
		tmp = Float64(Float64(Float64(eps * 5.0) * (x ^ 4.0)) + Float64(x * Float64(10.0 * Float64(x * Float64(Float64(eps * eps) * Float64(x + eps))))));
	else
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5.2e-38) || ~((x <= 4.7e-60)))
		tmp = ((eps * 5.0) * (x ^ 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	else
		tmp = (eps ^ 5.0) + (x * (5.0 * (eps ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5.2e-38], N[Not[LessEqual[x, 4.7e-60]], $MachinePrecision]], N[(N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000022e-38 or 4.7e-60 < x

   1. Initial program 34.2%

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

   2. Taylor expanded in x around inf 96.5%

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

      1. fma-def 96.5%

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

      2. distribute-lft1-in 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 96.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 96.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 96.5%

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

   5. Taylor expanded in x around 0 96.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10 + \color{blue}{\left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10}\right) \]

      4. unpow2 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      5. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(x \cdot x\right) \cdot {\varepsilon}^{3}\right)} \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      6. associate-*r* 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10\right)} + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      7. unpow2 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*r* 96.5%

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

      10. *-commutative 96.5%

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

      11. cube-mult 96.5%

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

      12. associate-*l* 96.5%

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

      13. *-commutative 96.5%

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

      14. distribute-lft-in 96.5%

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

      15. fma-udef 96.5%

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

      16. associate-*l* 96.5%

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

   7. Simplified 96.5%

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

      1. fma-udef 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*l* 96.5%

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

      4. cube-mult 96.5%

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

      5. distribute-rgt-out 96.5%

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

   9. Applied egg-rr 96.5%

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

   if -5.20000000000000022e-38 < x < 4.7e-60

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7.2e-38) (not (<= x 4.7e-60)))
   (+
    (* (* eps 5.0) (pow x 4.0))
    (* x (* 10.0 (* x (* (* eps eps) (+ x eps))))))
   (- (pow (+ x eps) 5.0) (pow x 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7.2e-38) || !(x <= 4.7e-60)) {
		tmp = ((eps * 5.0) * pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	} else {
		tmp = pow((x + eps), 5.0) - pow(x, 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 <= (-7.2d-38)) .or. (.not. (x <= 4.7d-60))) then
        tmp = ((eps * 5.0d0) * (x ** 4.0d0)) + (x * (10.0d0 * (x * ((eps * eps) * (x + eps)))))
    else
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -7.2e-38) || !(x <= 4.7e-60)) {
		tmp = ((eps * 5.0) * Math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	} else {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -7.2e-38) or not (x <= 4.7e-60):
		tmp = ((eps * 5.0) * math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))))
	else:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -7.2e-38) || !(x <= 4.7e-60))
		tmp = Float64(Float64(Float64(eps * 5.0) * (x ^ 4.0)) + Float64(x * Float64(10.0 * Float64(x * Float64(Float64(eps * eps) * Float64(x + eps))))));
	else
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -7.2e-38) || ~((x <= 4.7e-60)))
		tmp = ((eps * 5.0) * (x ^ 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	else
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -7.2e-38], N[Not[LessEqual[x, 4.7e-60]], $MachinePrecision]], N[(N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2000000000000001e-38 or 4.7e-60 < x

   1. Initial program 34.2%

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

   2. Taylor expanded in x around inf 96.5%

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

      1. fma-def 96.5%

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

      2. distribute-lft1-in 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 96.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 96.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 96.5%

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

   5. Taylor expanded in x around 0 96.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10 + \color{blue}{\left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10}\right) \]

      4. unpow2 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      5. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(x \cdot x\right) \cdot {\varepsilon}^{3}\right)} \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      6. associate-*r* 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10\right)} + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      7. unpow2 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*r* 96.5%

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

      10. *-commutative 96.5%

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

      11. cube-mult 96.5%

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

      12. associate-*l* 96.5%

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

      13. *-commutative 96.5%

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

      14. distribute-lft-in 96.5%

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

      15. fma-udef 96.5%

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

      16. associate-*l* 96.5%

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

   7. Simplified 96.5%

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

      1. fma-udef 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*l* 96.5%

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

      4. cube-mult 96.5%

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

      5. distribute-rgt-out 96.5%

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

   9. Applied egg-rr 96.5%

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

   if -7.2000000000000001e-38 < x < 4.7e-60

   1. Initial program 99.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7.2e-38) (not (<= x 4.7e-60)))
   (+
    (* (* eps 5.0) (pow x 4.0))
    (* x (* 10.0 (* x (* (* eps eps) (+ x eps))))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7.2e-38) || !(x <= 4.7e-60)) {
		tmp = ((eps * 5.0) * pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	} 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 <= (-7.2d-38)) .or. (.not. (x <= 4.7d-60))) then
        tmp = ((eps * 5.0d0) * (x ** 4.0d0)) + (x * (10.0d0 * (x * ((eps * eps) * (x + eps)))))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -7.2e-38) || !(x <= 4.7e-60)) {
		tmp = ((eps * 5.0) * Math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -7.2e-38) or not (x <= 4.7e-60):
		tmp = ((eps * 5.0) * math.pow(x, 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -7.2e-38) || !(x <= 4.7e-60))
		tmp = Float64(Float64(Float64(eps * 5.0) * (x ^ 4.0)) + Float64(x * Float64(10.0 * Float64(x * Float64(Float64(eps * eps) * Float64(x + eps))))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -7.2e-38) || ~((x <= 4.7e-60)))
		tmp = ((eps * 5.0) * (x ^ 4.0)) + (x * (10.0 * (x * ((eps * eps) * (x + eps)))));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -7.2e-38], N[Not[LessEqual[x, 4.7e-60]], $MachinePrecision]], N[(N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2000000000000001e-38 or 4.7e-60 < x

   1. Initial program 34.2%

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

   2. Taylor expanded in x around inf 96.5%

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

      1. fma-def 96.5%

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

      2. distribute-lft1-in 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 96.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 96.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 96.5%

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

   5. Taylor expanded in x around 0 96.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot {x}^{2}\right) \cdot 10 + \color{blue}{\left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10}\right) \]

      4. unpow2 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      5. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(x \cdot x\right) \cdot {\varepsilon}^{3}\right)} \cdot 10 + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      6. associate-*r* 96.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10\right)} + \left({\varepsilon}^{2} \cdot {x}^{3}\right) \cdot 10\right) \]

      7. unpow2 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*r* 96.5%

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

      10. *-commutative 96.5%

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

      11. cube-mult 96.5%

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

      12. associate-*l* 96.5%

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

      13. *-commutative 96.5%

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

      14. distribute-lft-in 96.5%

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

      15. fma-udef 96.5%

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

      16. associate-*l* 96.5%

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

   7. Simplified 96.5%

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

      1. fma-udef 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*l* 96.5%

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

      4. cube-mult 96.5%

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

      5. distribute-rgt-out 96.5%

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

   9. Applied egg-rr 96.5%

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

   if -7.2000000000000001e-38 < x < 4.7e-60

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + x \cdot \left(10 \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.2e-38) (not (<= x 4.7e-60)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e-38) || !(x <= 4.7e-60)) {
		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 <= (-5.2d-38)) .or. (.not. (x <= 4.7d-60))) 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 <= -5.2e-38) || !(x <= 4.7e-60)) {
		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 <= -5.2e-38) or not (x <= 4.7e-60):
		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 <= -5.2e-38) || !(x <= 4.7e-60))
		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 <= -5.2e-38) || ~((x <= 4.7e-60)))
		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, -5.2e-38], N[Not[LessEqual[x, 4.7e-60]], $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 < -5.20000000000000022e-38 or 4.7e-60 < x

   1. Initial program 34.2%

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

   2. Taylor expanded in x around inf 93.0%

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

      1. distribute-lft1-in 93.0%

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

      2. metadata-eval 93.0%

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

      3. associate-*l* 92.9%

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

   4. Simplified 92.9%

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

   if -5.20000000000000022e-38 < x < 4.7e-60

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 7: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-38)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 4.7e-60) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-38) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 4.7e-60) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.2d-38)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 4.7d-60) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-38) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 4.7e-60) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e-38:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 4.7e-60:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-38)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 4.7e-60)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e-38)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 4.7e-60)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-38], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.7e-60], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000022e-38

   1. Initial program 25.3%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. distribute-lft1-in 98.7%

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

      2. metadata-eval 98.7%

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

      3. associate-*l* 98.7%

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

   4. Simplified 98.7%

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

   if -5.20000000000000022e-38 < x < 4.7e-60

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

   if 4.7e-60 < x

   1. Initial program 39.2%

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

   2. Taylor expanded in eps around 0 89.9%

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

      1. *-un-lft-identity 89.9%

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

      2. distribute-rgt-out 89.9%

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

      3. metadata-eval 89.9%

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

   4. Applied egg-rr 89.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 8: 97.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot 5\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.2e-38) (not (<= x 4.7e-60)))
   (* eps (* (* x (* x 5.0)) (* x x)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e-38) || !(x <= 4.7e-60)) {
		tmp = eps * ((x * (x * 5.0)) * (x * x));
	} 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 <= (-5.2d-38)) .or. (.not. (x <= 4.7d-60))) then
        tmp = eps * ((x * (x * 5.0d0)) * (x * x))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e-38) || !(x <= 4.7e-60)) {
		tmp = eps * ((x * (x * 5.0)) * (x * x));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5.2e-38) or not (x <= 4.7e-60):
		tmp = eps * ((x * (x * 5.0)) * (x * x))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5.2e-38) || !(x <= 4.7e-60))
		tmp = Float64(eps * Float64(Float64(x * Float64(x * 5.0)) * Float64(x * x)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5.2e-38) || ~((x <= 4.7e-60)))
		tmp = eps * ((x * (x * 5.0)) * (x * x));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5.2e-38], N[Not[LessEqual[x, 4.7e-60]], $MachinePrecision]], N[(eps * N[(N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000022e-38 or 4.7e-60 < x

   1. Initial program 34.2%

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

   2. Taylor expanded in eps around 0 93.1%

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

      1. distribute-lft1-in 93.1%

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

      2. metadata-eval 93.1%

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

      3. metadata-eval 93.1%

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

      4. pow-sqr 92.8%

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

      5. pow2 92.8%

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

      6. pow2 92.8%

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

      7. associate-*r* 92.7%

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

   4. Applied egg-rr 92.7%

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

   5. Taylor expanded in x around 0 92.7%

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

      1. unpow2 92.7%

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

      2. *-commutative 92.7%

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

      3. associate-*r* 92.8%

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

   7. Simplified 92.8%

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

   if -5.20000000000000022e-38 < x < 4.7e-60

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-38} \lor \neg \left(x \leq 4.7 \cdot 10^{-60}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot 5\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 9: 82.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in x around inf 80.3%

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

   1. distribute-lft1-in 80.3%

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

   2. metadata-eval 80.3%

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

   3. associate-*l* 80.3%

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

4. Simplified 80.3%

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

   1. add-sqr-sqrt 74.1%

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

   2. pow2 74.1%

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

   3. *-commutative 74.1%

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

   4. sqrt-prod 38.4%

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

   5. sqrt-pow1 38.4%

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

   6. metadata-eval 38.4%

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

   7. pow2 38.4%

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

6. Applied egg-rr 38.4%

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

   1. unpow2 38.4%

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

   2. *-commutative 38.4%

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

   3. *-commutative 38.4%

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

   4. swap-sqr 38.4%

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

   5. add-sqr-sqrt 80.3%

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

   6. associate-*r* 80.3%

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

8. Applied egg-rr 80.3%

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

9. Final simplification 80.3%

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

Alternative 10: 82.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in eps around 0 80.4%

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

   1. distribute-lft1-in 80.4%

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

   2. metadata-eval 80.4%

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

   3. metadata-eval 80.4%

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

   4. pow-sqr 80.3%

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

   5. pow2 80.3%

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

   6. pow2 80.3%

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

   7. associate-*r* 80.3%

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

4. Applied egg-rr 80.3%

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

5. Final simplification 80.3%

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

Alternative 11: 82.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in eps around 0 80.4%

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

   1. distribute-lft1-in 80.4%

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

   2. metadata-eval 80.4%

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

   3. metadata-eval 80.4%

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

   4. pow-sqr 80.3%

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

   5. pow2 80.3%

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

   6. pow2 80.3%

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

   7. associate-*r* 80.3%

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

4. Applied egg-rr 80.3%

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

5. Taylor expanded in x around 0 80.3%

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

   1. unpow2 80.3%

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

   2. *-commutative 80.3%

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

   3. associate-*r* 80.3%

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

7. Simplified 80.3%

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

8. Final simplification 80.3%

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

Reproduce

herbie shell --seed 2023192 
(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)))