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

Percentage Accurate: 89.0% → 98.4%

Time: 10.8s

Alternatives: 17

Speedup: 5.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.1e-52)
   (* eps (fma (* x (* x (* x x))) 5.0 (* x (* eps (* x (* x 10.0))))))
   (if (<= x 4.9e-54)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (*
      eps
      (*
       x
       (fma
        x
        (fma x (* x 5.0) (* 10.0 (* eps (+ x eps))))
        (* 5.0 (* eps (* eps eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.1e-52) {
		tmp = eps * fma((x * (x * (x * x))), 5.0, (x * (eps * (x * (x * 10.0)))));
	} else if (x <= 4.9e-54) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * (x * fma(x, fma(x, (x * 5.0), (10.0 * (eps * (x + eps)))), (5.0 * (eps * (eps * eps)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.1e-52)
		tmp = Float64(eps * fma(Float64(x * Float64(x * Float64(x * x))), 5.0, Float64(x * Float64(eps * Float64(x * Float64(x * 10.0))))));
	elseif (x <= 4.9e-54)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(eps * Float64(x * fma(x, fma(x, Float64(x * 5.0), Float64(10.0 * Float64(eps * Float64(x + eps)))), Float64(5.0 * Float64(eps * Float64(eps * eps))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.1e-52], N[(eps * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(x * N[(eps * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(x * N[(x * N[(x * 5.0), $MachinePrecision] + N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.0999999999999999e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   if -6.0999999999999999e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 95.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.9e-52)
   (* eps (fma (* x (* x (* x x))) 5.0 (* x (* eps (* x (* x 10.0))))))
   (if (<= x 4.9e-54)
     (* (pow eps 5.0) (/ (fma x 5.0 eps) eps))
     (*
      eps
      (*
       x
       (fma
        x
        (fma x (* x 5.0) (* 10.0 (* eps (+ x eps))))
        (* 5.0 (* eps (* eps eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * fma((x * (x * (x * x))), 5.0, (x * (eps * (x * (x * 10.0)))));
	} else if (x <= 4.9e-54) {
		tmp = pow(eps, 5.0) * (fma(x, 5.0, eps) / eps);
	} else {
		tmp = eps * (x * fma(x, fma(x, (x * 5.0), (10.0 * (eps * (x + eps)))), (5.0 * (eps * (eps * eps)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * fma(Float64(x * Float64(x * Float64(x * x))), 5.0, Float64(x * Float64(eps * Float64(x * Float64(x * 10.0))))));
	elseif (x <= 4.9e-54)
		tmp = Float64((eps ^ 5.0) * Float64(fma(x, 5.0, eps) / eps));
	else
		tmp = Float64(eps * Float64(x * fma(x, fma(x, Float64(x * 5.0), Float64(10.0 * Float64(eps * Float64(x + eps)))), Float64(5.0 * Float64(eps * Float64(eps * eps))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.9e-52], N[(eps * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(x * N[(eps * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(N[(x * 5.0 + eps), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(x * N[(x * N[(x * 5.0), $MachinePrecision] + N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.7%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 95.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.9e-52)
   (* eps (fma (* x (* x (* x x))) 5.0 (* x (* eps (* x (* x 10.0))))))
   (if (<= x 4.2e-54)
     (pow eps 5.0)
     (*
      eps
      (*
       x
       (fma
        x
        (fma x (* x 5.0) (* 10.0 (* eps (+ x eps))))
        (* 5.0 (* eps (* eps eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * fma((x * (x * (x * x))), 5.0, (x * (eps * (x * (x * 10.0)))));
	} else if (x <= 4.2e-54) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (x * fma(x, fma(x, (x * 5.0), (10.0 * (eps * (x + eps)))), (5.0 * (eps * (eps * eps)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * fma(Float64(x * Float64(x * Float64(x * x))), 5.0, Float64(x * Float64(eps * Float64(x * Float64(x * 10.0))))));
	elseif (x <= 4.2e-54)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(x * fma(x, fma(x, Float64(x * 5.0), Float64(10.0 * Float64(eps * Float64(x + eps)))), Float64(5.0 * Float64(eps * Float64(eps * eps))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.9e-52], N[(eps * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(x * N[(eps * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-54], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(x * N[(x * N[(x * N[(x * 5.0), $MachinePrecision] + N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   if -3.90000000000000018e-52 < x < 4.2e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 4.2e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 95.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps eps))))
   (if (<= x -3.9e-52)
     (* eps (fma (* x (* x (* x x))) 5.0 (* x (* eps (* x (* x 10.0))))))
     (if (<= x 4.9e-54)
       (* (/ (fma x 5.0 eps) eps) (* (* eps eps) t_0))
       (*
        eps
        (*
         x
         (fma x (fma x (* x 5.0) (* 10.0 (* eps (+ x eps)))) (* 5.0 t_0))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * fma((x * (x * (x * x))), 5.0, (x * (eps * (x * (x * 10.0)))));
	} else if (x <= 4.9e-54) {
		tmp = (fma(x, 5.0, eps) / eps) * ((eps * eps) * t_0);
	} else {
		tmp = eps * (x * fma(x, fma(x, (x * 5.0), (10.0 * (eps * (x + eps)))), (5.0 * t_0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * fma(Float64(x * Float64(x * Float64(x * x))), 5.0, Float64(x * Float64(eps * Float64(x * Float64(x * 10.0))))));
	elseif (x <= 4.9e-54)
		tmp = Float64(Float64(fma(x, 5.0, eps) / eps) * Float64(Float64(eps * eps) * t_0));
	else
		tmp = Float64(eps * Float64(x * fma(x, fma(x, Float64(x * 5.0), Float64(10.0 * Float64(eps * Float64(x + eps)))), Float64(5.0 * t_0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(x * N[(eps * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(N[(N[(x * 5.0 + eps), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(x * N[(x * N[(x * 5.0), $MachinePrecision] + N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 95.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 5, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right), 5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot t\_0, 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 10 \cdot \left(\left(x \cdot x\right) \cdot \left(x + \varepsilon\right)\right), t\_0 \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.9e-52)
     (* eps (fma (* x t_0) 5.0 (* x (* eps (* x (* x 10.0))))))
     (if (<= x 4.2e-54)
       (* (/ (fma x 5.0 eps) eps) (* (* eps eps) (* eps (* eps eps))))
       (* eps (fma eps (* 10.0 (* (* x x) (+ x eps))) (* t_0 (* x 5.0))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * fma((x * t_0), 5.0, (x * (eps * (x * (x * 10.0)))));
	} else if (x <= 4.2e-54) {
		tmp = (fma(x, 5.0, eps) / eps) * ((eps * eps) * (eps * (eps * eps)));
	} else {
		tmp = eps * fma(eps, (10.0 * ((x * x) * (x + eps))), (t_0 * (x * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * fma(Float64(x * t_0), 5.0, Float64(x * Float64(eps * Float64(x * Float64(x * 10.0))))));
	elseif (x <= 4.2e-54)
		tmp = Float64(Float64(fma(x, 5.0, eps) / eps) * Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(eps * fma(eps, Float64(10.0 * Float64(Float64(x * x) * Float64(x + eps))), Float64(t_0 * Float64(x * 5.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(N[(x * t$95$0), $MachinePrecision] * 5.0 + N[(x * N[(eps * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-54], N[(N[(N[(x * 5.0 + eps), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * N[(10.0 * N[(N[(x * x), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   if -3.90000000000000018e-52 < x < 4.2e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.6%

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

   if 4.2e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 95.5%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 95.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 10 \cdot \left(\left(x \cdot x\right) \cdot \left(x + \varepsilon\right)\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (* eps (fma (* x (* x (* x x))) 5.0 (* x (* eps (* x (* x 10.0))))))))
   (if (<= x -3.9e-52)
     t_0
     (if (<= x 4.9e-54)
       (* (/ (fma x 5.0 eps) eps) (* (* eps eps) (* eps (* eps eps))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * fma((x * (x * (x * x))), 5.0, (x * (eps * (x * (x * 10.0)))));
	double tmp;
	if (x <= -3.9e-52) {
		tmp = t_0;
	} else if (x <= 4.9e-54) {
		tmp = (fma(x, 5.0, eps) / eps) * ((eps * eps) * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * fma(Float64(x * Float64(x * Float64(x * x))), 5.0, Float64(x * Float64(eps * Float64(x * Float64(x * 10.0))))))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = t_0;
	elseif (x <= 4.9e-54)
		tmp = Float64(Float64(fma(x, 5.0, eps) / eps) * Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(x * N[(eps * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], t$95$0, If[LessEqual[x, 4.9e-54], N[(N[(N[(x * 5.0 + eps), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.90000000000000018e-52 or 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.7%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(10, \varepsilon \cdot \varepsilon, x \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.9e-52)
     (* eps (* t_0 (fma 10.0 eps (* x 5.0))))
     (if (<= x 4.9e-54)
       (* (/ (fma x 5.0 eps) eps) (* (* eps eps) (* eps (* eps eps))))
       (* t_0 (fma 10.0 (* eps eps) (* x (* eps 5.0))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * fma(10.0, eps, (x * 5.0)));
	} else if (x <= 4.9e-54) {
		tmp = (fma(x, 5.0, eps) / eps) * ((eps * eps) * (eps * (eps * eps)));
	} else {
		tmp = t_0 * fma(10.0, (eps * eps), (x * (eps * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * Float64(t_0 * fma(10.0, eps, Float64(x * 5.0))));
	elseif (x <= 4.9e-54)
		tmp = Float64(Float64(fma(x, 5.0, eps) / eps) * Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * fma(10.0, Float64(eps * eps), Float64(x * Float64(eps * 5.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(t$95$0 * N[(10.0 * eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(N[(N[(x * 5.0 + eps), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(10.0 * N[(eps * eps), $MachinePrecision] + N[(x * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.7%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 5, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(10, \varepsilon \cdot \varepsilon, x \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(10, \varepsilon \cdot \varepsilon, x \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.9e-52)
     (* eps (* t_0 (fma 10.0 eps (* x 5.0))))
     (if (<= x 4.9e-54)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (fma 10.0 (* eps eps) (* x (* eps 5.0))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * fma(10.0, eps, (x * 5.0)));
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * fma(10.0, (eps * eps), (x * (eps * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * Float64(t_0 * fma(10.0, eps, Float64(x * 5.0))));
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * fma(10.0, Float64(eps * eps), Float64(x * Float64(eps * 5.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(t$95$0 * N[(10.0 * eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(10.0 * N[(eps * eps), $MachinePrecision] + N[(x * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.7%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(10, \varepsilon \cdot \varepsilon, x \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma 10.0 eps (* x 5.0))))
   (if (<= x -3.9e-52)
     (* eps (* (* x (* x x)) t_0))
     (if (<= x 4.9e-54)
       (* eps (* eps (* eps (* eps eps))))
       (* eps (* x (* (* x x) t_0)))))))

C

double code(double x, double eps) {
	double t_0 = fma(10.0, eps, (x * 5.0));
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * ((x * (x * x)) * t_0);
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (x * ((x * x) * t_0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(10.0, eps, Float64(x * 5.0))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * Float64(Float64(x * Float64(x * x)) * t_0));
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(eps * Float64(x * Float64(Float64(x * x) * t_0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(10.0 * eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.7%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 94.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (fma 10.0 eps (* x 5.0))))
   (if (<= x -3.9e-52)
     (* eps (* t_0 t_1))
     (if (<= x 4.9e-54)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (* eps t_1))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = fma(10.0, eps, (x * 5.0));
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * t_1);
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * t_1);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	t_1 = fma(10.0, eps, Float64(x * 5.0))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * Float64(t_0 * t_1));
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * Float64(eps * t_1));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(10.0 * eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.7%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 94.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x (* x x)) (* eps (fma 10.0 eps (* x 5.0))))))
   (if (<= x -3.9e-52)
     t_0
     (if (<= x 4.9e-54) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = (x * (x * x)) * (eps * fma(10.0, eps, (x * 5.0)));
	double tmp;
	if (x <= -3.9e-52) {
		tmp = t_0;
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * Float64(x * x)) * Float64(eps * fma(10.0, eps, Float64(x * 5.0))))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = t_0;
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(10.0 * eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], t$95$0, If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.90000000000000018e-52 or 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 96.3%

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

   7. Applied rewrites 96.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.4%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(x \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.9e-52)
     (* eps (* t_0 (* x 5.0)))
     (if (<= x 4.9e-54)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (* x (* eps 5.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * (x * 5.0));
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (x * (eps * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-3.9d-52)) then
        tmp = eps * (t_0 * (x * 5.0d0))
    else if (x <= 4.9d-54) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0 * (x * (eps * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * (x * 5.0));
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (x * (eps * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -3.9e-52:
		tmp = eps * (t_0 * (x * 5.0))
	elif x <= 4.9e-54:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0 * (x * (eps * 5.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * Float64(t_0 * Float64(x * 5.0)));
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * Float64(x * Float64(eps * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -3.9e-52)
		tmp = eps * (t_0 * (x * 5.0));
	elseif (x <= 4.9e-54)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0 * (x * (eps * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(t$95$0 * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.1%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 92.5%

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

   12. Applied rewrites 92.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\_0\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.9e-52)
     (* eps (* t_0 (* x 5.0)))
     (if (<= x 4.9e-54)
       (* eps (* eps (* eps (* eps eps))))
       (* (* x t_0) (* eps 5.0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * (x * 5.0));
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = (x * t_0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-3.9d-52)) then
        tmp = eps * (t_0 * (x * 5.0d0))
    else if (x <= 4.9d-54) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = (x * t_0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * (x * 5.0));
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = (x * t_0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -3.9e-52:
		tmp = eps * (t_0 * (x * 5.0))
	elif x <= 4.9e-54:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = (x * t_0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * Float64(t_0 * Float64(x * 5.0)));
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(Float64(x * t_0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -3.9e-52)
		tmp = eps * (t_0 * (x * 5.0));
	elseif (x <= 4.9e-54)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = (x * t_0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(t$95$0 * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.1%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 92.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.9e-52)
     (* eps (* t_0 (* x 5.0)))
     (if (<= x 4.9e-54)
       (* eps (* eps (* eps (* eps eps))))
       (* 5.0 (* eps (* x t_0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * (x * 5.0));
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = 5.0 * (eps * (x * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-3.9d-52)) then
        tmp = eps * (t_0 * (x * 5.0d0))
    else if (x <= 4.9d-54) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = 5.0d0 * (eps * (x * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.9e-52) {
		tmp = eps * (t_0 * (x * 5.0));
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = 5.0 * (eps * (x * t_0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -3.9e-52:
		tmp = eps * (t_0 * (x * 5.0))
	elif x <= 4.9e-54:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = 5.0 * (eps * (x * t_0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = Float64(eps * Float64(t_0 * Float64(x * 5.0)));
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(5.0 * Float64(eps * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -3.9e-52)
		tmp = eps * (t_0 * (x * 5.0));
	elseif (x <= 4.9e-54)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = 5.0 * (eps * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], N[(eps * N[(t$95$0 * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000018e-52

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.1%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 92.3

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

   10. Applied rewrites 92.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (* eps (* x (* x (* x x)))))))
   (if (<= x -3.9e-52)
     t_0
     (if (<= x 4.9e-54) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * (eps * (x * (x * (x * x))));
	double tmp;
	if (x <= -3.9e-52) {
		tmp = t_0;
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 5.0d0 * (eps * (x * (x * (x * x))))
    if (x <= (-3.9d-52)) then
        tmp = t_0
    else if (x <= 4.9d-54) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 5.0 * (eps * (x * (x * (x * x))));
	double tmp;
	if (x <= -3.9e-52) {
		tmp = t_0;
	} else if (x <= 4.9e-54) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 5.0 * (eps * (x * (x * (x * x))))
	tmp = 0
	if x <= -3.9e-52:
		tmp = t_0
	elif x <= 4.9e-54:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(5.0 * Float64(eps * Float64(x * Float64(x * Float64(x * x)))))
	tmp = 0.0
	if (x <= -3.9e-52)
		tmp = t_0;
	elseif (x <= 4.9e-54)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 5.0 * (eps * (x * (x * (x * x))));
	tmp = 0.0;
	if (x <= -3.9e-52)
		tmp = t_0;
	elseif (x <= 4.9e-54)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[(eps * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e-52], t$95$0, If[LessEqual[x, 4.9e-54], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.90000000000000018e-52 or 4.90000000000000021e-54 < x

   1. Initial program 29.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.6

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

   10. Applied rewrites 93.6%

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

   if -3.90000000000000018e-52 < x < 4.90000000000000021e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-52}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.9% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

3. Taylor expanded in x around 0

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

   1. lower-pow.f64 85.1

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

5. Applied rewrites 85.1%

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

7. Applied rewrites 85.0%

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

9. Applied rewrites 85.0%

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

10. Final simplification 85.0%

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

Alternative 17: 87.9% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

3. Taylor expanded in x around 0

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

   1. lower-pow.f64 85.1

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

5. Applied rewrites 85.1%

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

7. Applied rewrites 85.0%

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

8. Final simplification 85.0%

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

Reproduce

herbie shell --seed 2024234 
(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)))