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

Percentage Accurate: 88.3% → 98.1%

Time: 12.6s

Alternatives: 8

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-60} \lor \neg \left(x \leq 9.5 \cdot 10^{-52}\right):\\ \;\;\;\;{x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 6\right)\right)\right) + \left({x}^{3} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + {x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.2e-60) (not (<= x 9.5e-52)))
   (+
    (* (pow x 2.0) (+ (* 4.0 (pow eps 3.0)) (* eps (* eps (* eps 6.0)))))
    (+
     (* (pow x 3.0) (* eps (* eps 10.0)))
     (* (pow x 4.0) (+ eps (* 4.0 eps)))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.2e-60) || !(x <= 9.5e-52)) {
		tmp = (pow(x, 2.0) * ((4.0 * pow(eps, 3.0)) + (eps * (eps * (eps * 6.0))))) + ((pow(x, 3.0) * (eps * (eps * 10.0))) + (pow(x, 4.0) * (eps + (4.0 * eps))));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-4.2d-60)) .or. (.not. (x <= 9.5d-52))) then
        tmp = ((x ** 2.0d0) * ((4.0d0 * (eps ** 3.0d0)) + (eps * (eps * (eps * 6.0d0))))) + (((x ** 3.0d0) * (eps * (eps * 10.0d0))) + ((x ** 4.0d0) * (eps + (4.0d0 * eps))))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -4.2e-60) || !(x <= 9.5e-52)) {
		tmp = (Math.pow(x, 2.0) * ((4.0 * Math.pow(eps, 3.0)) + (eps * (eps * (eps * 6.0))))) + ((Math.pow(x, 3.0) * (eps * (eps * 10.0))) + (Math.pow(x, 4.0) * (eps + (4.0 * eps))));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -4.2e-60) or not (x <= 9.5e-52):
		tmp = (math.pow(x, 2.0) * ((4.0 * math.pow(eps, 3.0)) + (eps * (eps * (eps * 6.0))))) + ((math.pow(x, 3.0) * (eps * (eps * 10.0))) + (math.pow(x, 4.0) * (eps + (4.0 * eps))))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -4.2e-60) || !(x <= 9.5e-52))
		tmp = Float64(Float64((x ^ 2.0) * Float64(Float64(4.0 * (eps ^ 3.0)) + Float64(eps * Float64(eps * Float64(eps * 6.0))))) + Float64(Float64((x ^ 3.0) * Float64(eps * Float64(eps * 10.0))) + Float64((x ^ 4.0) * Float64(eps + Float64(4.0 * eps)))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -4.2e-60) || ~((x <= 9.5e-52)))
		tmp = ((x ^ 2.0) * ((4.0 * (eps ^ 3.0)) + (eps * (eps * (eps * 6.0))))) + (((x ^ 3.0) * (eps * (eps * 10.0))) + ((x ^ 4.0) * (eps + (4.0 * eps))));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -4.2e-60], N[Not[LessEqual[x, 9.5e-52]], $MachinePrecision]], N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(4.0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(eps * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps + N[(4.0 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.19999999999999982e-60 or 9.50000000000000007e-52 < x

   1. Initial program 41.3%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. distribute-rgt-out 98.5%

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

      2. metadata-eval 98.5%

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

      3. *-commutative 98.5%

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

      4. unpow2 98.5%

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

      5. associate-*r* 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. distribute-rgt-out 98.5%

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

      2. unpow2 98.5%

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

      3. associate-*l* 98.5%

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

      4. metadata-eval 98.5%

         \[\leadsto {x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{6}\right)\right)\right) + \left({x}^{3} \cdot \left(\left(10 \cdot \varepsilon\right) \cdot \varepsilon\right) + {x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) \]

   6. Applied egg-rr 98.5%

      \[\leadsto {x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot 6\right)\right)}\right) + \left({x}^{3} \cdot \left(\left(10 \cdot \varepsilon\right) \cdot \varepsilon\right) + {x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) \]

   if -4.19999999999999982e-60 < x < 9.50000000000000007e-52

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-60} \lor \neg \left(x \leq 9.5 \cdot 10^{-52}\right):\\ \;\;\;\;{x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 6\right)\right)\right) + \left({x}^{3} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + {x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 2: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-256}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-256)
     (pow eps 5.0)
     (if (<= t_0 0.0) (* 5.0 (* eps (pow x 4.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-256) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} 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 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-5d-256)) then
        tmp = eps ** 5.0d0
    else if (t_0 <= 0.0d0) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-256) {
		tmp = Math.pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -5e-256:
		tmp = math.pow(eps, 5.0)
	elif t_0 <= 0.0:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-256)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -5e-256)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-256], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -5e-256

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if -5e-256 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in eps around 0 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 92.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-256}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot {x}^{4}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \left(10 \cdot {\varepsilon}^{2}\right) + 5 \cdot t_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-51}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, t_0, 10 \cdot \left({x}^{3} \cdot {\varepsilon}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (pow x 4.0))))
   (if (<= x -2.7e-58)
     (+ (* (pow x 3.0) (* 10.0 (pow eps 2.0))) (* 5.0 t_0))
     (if (<= x 1.15e-51)
       (pow eps 5.0)
       (fma 5.0 t_0 (* 10.0 (* (pow x 3.0) (pow eps 2.0))))))))

C

double code(double x, double eps) {
	double t_0 = eps * pow(x, 4.0);
	double tmp;
	if (x <= -2.7e-58) {
		tmp = (pow(x, 3.0) * (10.0 * pow(eps, 2.0))) + (5.0 * t_0);
	} else if (x <= 1.15e-51) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = fma(5.0, t_0, (10.0 * (pow(x, 3.0) * pow(eps, 2.0))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * (x ^ 4.0))
	tmp = 0.0
	if (x <= -2.7e-58)
		tmp = Float64(Float64((x ^ 3.0) * Float64(10.0 * (eps ^ 2.0))) + Float64(5.0 * t_0));
	elseif (x <= 1.15e-51)
		tmp = eps ^ 5.0;
	else
		tmp = fma(5.0, t_0, Float64(10.0 * Float64((x ^ 3.0) * (eps ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-58], N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(10.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-51], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * t$95$0 + N[(10.0 * N[(N[Power[x, 3.0], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6999999999999999e-58

   1. Initial program 44.1%

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

   2. Taylor expanded in x around inf 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-def 99.3%

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

      3. distribute-rgt1-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. *-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-out 99.3%

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

      8. metadata-eval 99.3%

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

      9. associate-*l* 99.3%

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

   4. Simplified 99.3%

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

      1. fma-udef 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-*r* 99.3%

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

      4. *-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*r* 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*l* 99.3%

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

   6. Applied egg-rr 99.3%

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

   if -2.6999999999999999e-58 < x < 1.15000000000000001e-51

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 1.15000000000000001e-51 < x

   1. Initial program 38.9%

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

      1. add-cbrt-cube 12.0%

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

      2. unpow3 8.7%

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

      3. pow-pow 12.0%

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

      4. metadata-eval 12.0%

         \[\leadsto \sqrt[3]{{\left(x + \varepsilon\right)}^{\color{blue}{15}}} - {x}^{5} \]

   3. Applied egg-rr 12.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + \varepsilon\right)}^{15}}} - {x}^{5} \]

   4. Taylor expanded in x around inf 97.2%

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

      1. +-commutative 97.2%

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

      2. distribute-rgt-out 97.2%

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

      3. metadata-eval 97.2%

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

      4. *-commutative 97.2%

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

      5. *-commutative 97.2%

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

      6. distribute-rgt1-in 97.2%

         \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} + {x}^{3} \cdot \left(10 \cdot {\varepsilon}^{2}\right) \]

      7. metadata-eval 97.2%

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

      8. associate-*r* 97.3%

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

      9. fma-def 97.3%

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

      10. *-commutative 97.3%

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

      11. associate-*r* 97.3%

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

   6. Simplified 97.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \left(10 \cdot {\varepsilon}^{2}\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-51}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon \cdot {x}^{4}, 10 \cdot \left({x}^{3} \cdot {\varepsilon}^{2}\right)\right)\\ \end{array} \]

Alternative 4: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58} \lor \neg \left(x \leq 10^{-51}\right):\\ \;\;\;\;{\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-58) (not (<= x 1e-51)))
   (+ (* (pow eps 2.0) (* (pow x 3.0) 10.0)) (* eps (* (pow x 4.0) 5.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-58) || !(x <= 1e-51)) {
		tmp = (pow(eps, 2.0) * (pow(x, 3.0) * 10.0)) + (eps * (pow(x, 4.0) * 5.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.7d-58)) .or. (.not. (x <= 1d-51))) then
        tmp = ((eps ** 2.0d0) * ((x ** 3.0d0) * 10.0d0)) + (eps * ((x ** 4.0d0) * 5.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-58) || !(x <= 1e-51)) {
		tmp = (Math.pow(eps, 2.0) * (Math.pow(x, 3.0) * 10.0)) + (eps * (Math.pow(x, 4.0) * 5.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.7e-58) or not (x <= 1e-51):
		tmp = (math.pow(eps, 2.0) * (math.pow(x, 3.0) * 10.0)) + (eps * (math.pow(x, 4.0) * 5.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.7e-58) || !(x <= 1e-51))
		tmp = Float64(Float64((eps ^ 2.0) * Float64((x ^ 3.0) * 10.0)) + Float64(eps * Float64((x ^ 4.0) * 5.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.7e-58) || ~((x <= 1e-51)))
		tmp = ((eps ^ 2.0) * ((x ^ 3.0) * 10.0)) + (eps * ((x ^ 4.0) * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.7e-58], N[Not[LessEqual[x, 1e-51]], $MachinePrecision]], N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6999999999999999e-58 or 1e-51 < x

   1. Initial program 41.3%

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

      1. add-cbrt-cube 13.0%

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

      2. unpow3 11.2%

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

      3. pow-pow 13.0%

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

      4. metadata-eval 13.0%

         \[\leadsto \sqrt[3]{{\left(x + \varepsilon\right)}^{\color{blue}{15}}} - {x}^{5} \]

   3. Applied egg-rr 13.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + \varepsilon\right)}^{15}}} - {x}^{5} \]

   4. Taylor expanded in x around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. distribute-rgt-out 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-commutative 98.2%

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

      5. *-commutative 98.2%

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

      6. distribute-rgt1-in 98.2%

         \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} + {x}^{3} \cdot \left(10 \cdot {\varepsilon}^{2}\right) \]

      7. metadata-eval 98.2%

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

      8. associate-*r* 98.2%

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

      9. fma-def 98.2%

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

      10. *-commutative 98.2%

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

      11. associate-*r* 98.2%

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

   6. Simplified 98.2%

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

      1. fma-udef 98.2%

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

      2. +-commutative 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*l* 98.2%

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

      5. *-commutative 98.2%

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

      6. associate-*r* 98.2%

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

      7. *-commutative 98.2%

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

   8. Applied egg-rr 98.2%

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

   if -2.6999999999999999e-58 < x < 1e-51

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58} \lor \neg \left(x \leq 10^{-51}\right):\\ \;\;\;\;{\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-58} \lor \neg \left(x \leq 1.1 \cdot 10^{-51}\right):\\ \;\;\;\;{x}^{3} \cdot \left(10 \cdot {\varepsilon}^{2}\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.1e-58) (not (<= x 1.1e-51)))
   (+ (* (pow x 3.0) (* 10.0 (pow eps 2.0))) (* 5.0 (* eps (pow x 4.0))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.1e-58) || !(x <= 1.1e-51)) {
		tmp = (pow(x, 3.0) * (10.0 * pow(eps, 2.0))) + (5.0 * (eps * pow(x, 4.0)));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.1d-58)) .or. (.not. (x <= 1.1d-51))) then
        tmp = ((x ** 3.0d0) * (10.0d0 * (eps ** 2.0d0))) + (5.0d0 * (eps * (x ** 4.0d0)))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.1e-58) || !(x <= 1.1e-51)) {
		tmp = (Math.pow(x, 3.0) * (10.0 * Math.pow(eps, 2.0))) + (5.0 * (eps * Math.pow(x, 4.0)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.1e-58) or not (x <= 1.1e-51):
		tmp = (math.pow(x, 3.0) * (10.0 * math.pow(eps, 2.0))) + (5.0 * (eps * math.pow(x, 4.0)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.1e-58) || !(x <= 1.1e-51))
		tmp = Float64(Float64((x ^ 3.0) * Float64(10.0 * (eps ^ 2.0))) + Float64(5.0 * Float64(eps * (x ^ 4.0))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.1e-58) || ~((x <= 1.1e-51)))
		tmp = ((x ^ 3.0) * (10.0 * (eps ^ 2.0))) + (5.0 * (eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.1e-58], N[Not[LessEqual[x, 1.1e-51]], $MachinePrecision]], N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(10.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999988e-58 or 1.1e-51 < x

   1. Initial program 41.3%

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

   2. Taylor expanded in x around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. fma-def 98.2%

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

      3. distribute-rgt1-in 98.2%

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

      4. metadata-eval 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-rgt-out 98.2%

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

      8. metadata-eval 98.2%

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

      9. associate-*l* 98.2%

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

   4. Simplified 98.2%

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

      1. fma-udef 98.2%

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

      2. +-commutative 98.2%

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

      3. associate-*r* 98.2%

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

      4. *-commutative 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. *-commutative 98.2%

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

      9. associate-*l* 98.2%

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

   6. Applied egg-rr 98.2%

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

   if -2.09999999999999988e-58 < x < 1.1e-51

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-58} \lor \neg \left(x \leq 1.1 \cdot 10^{-51}\right):\\ \;\;\;\;{x}^{3} \cdot \left(10 \cdot {\varepsilon}^{2}\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58} \lor \neg \left(x \leq 1.2 \cdot 10^{-51}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-58) (not (<= x 1.2e-51)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-58) || !(x <= 1.2e-51)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.7d-58)) .or. (.not. (x <= 1.2d-51))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-58) || !(x <= 1.2e-51)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.7e-58) or not (x <= 1.2e-51):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.7e-58) || !(x <= 1.2e-51))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.7e-58) || ~((x <= 1.2e-51)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.7e-58], N[Not[LessEqual[x, 1.2e-51]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6999999999999999e-58 or 1.2e-51 < x

   1. Initial program 41.3%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. *-commutative 95.7%

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

      2. distribute-rgt1-in 95.7%

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

      3. metadata-eval 95.7%

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

      4. *-commutative 95.7%

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

      5. associate-*r* 95.8%

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

   4. Simplified 95.8%

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

   5. Taylor expanded in eps around 0 95.8%

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

   if -2.6999999999999999e-58 < x < 1.2e-51

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58} \lor \neg \left(x \leq 1.2 \cdot 10^{-51}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-51}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-58)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 1.02e-51) (pow eps 5.0) (* eps (* (pow x 4.0) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-58) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.02e-51) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (pow(x, 4.0) * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-58)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.02d-51) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * ((x ** 4.0d0) * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-58) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.02e-51) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * (Math.pow(x, 4.0) * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-58:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.02e-51:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * (math.pow(x, 4.0) * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-58)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.02e-51)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-58)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.02e-51)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((x ^ 4.0) * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-58], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.02e-51], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6999999999999999e-58

   1. Initial program 44.1%

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

   2. Taylor expanded in x around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. distribute-rgt1-in 94.4%

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

      3. metadata-eval 94.4%

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

      4. *-commutative 94.4%

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

      5. associate-*r* 94.4%

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

   4. Simplified 94.4%

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

   5. Taylor expanded in eps around 0 94.4%

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

   if -2.6999999999999999e-58 < x < 1.01999999999999998e-51

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 1.01999999999999998e-51 < x

   1. Initial program 38.9%

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

   2. Taylor expanded in x around inf 96.8%

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

      1. *-commutative 96.8%

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

      2. distribute-rgt1-in 96.8%

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

      3. metadata-eval 96.8%

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

      4. *-commutative 96.8%

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

      5. associate-*r* 97.0%

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

   4. Simplified 97.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-51}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \]

Alternative 8: 87.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in x around 0 86.9%

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

3. Final simplification 86.9%

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

Reproduce

herbie shell --seed 2023301 
(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)))