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

Percentage Accurate: 89.1% → 98.0%

Time: 7.9s

Alternatives: 7

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% 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: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-{x}^{3}, {\varepsilon}^{2} \cdot -10, {x}^{2} \cdot \left(10 \cdot {\varepsilon}^{3}\right)\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4} + x \cdot {\varepsilon}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.4e-41)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 3.7e-59)
     (+ (* x (* 5.0 (pow eps 4.0))) (pow eps 5.0))
     (+
      (fma
       (- (pow x 3.0))
       (* (pow eps 2.0) -10.0)
       (* (pow x 2.0) (* 10.0 (pow eps 3.0))))
      (* 5.0 (+ (* eps (pow x 4.0)) (* x (pow eps 4.0))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-41) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 3.7e-59) {
		tmp = (x * (5.0 * pow(eps, 4.0))) + pow(eps, 5.0);
	} else {
		tmp = fma(-pow(x, 3.0), (pow(eps, 2.0) * -10.0), (pow(x, 2.0) * (10.0 * pow(eps, 3.0)))) + (5.0 * ((eps * pow(x, 4.0)) + (x * pow(eps, 4.0))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.4e-41)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 3.7e-59)
		tmp = Float64(Float64(x * Float64(5.0 * (eps ^ 4.0))) + (eps ^ 5.0));
	else
		tmp = Float64(fma(Float64(-(x ^ 3.0)), Float64((eps ^ 2.0) * -10.0), Float64((x ^ 2.0) * Float64(10.0 * (eps ^ 3.0)))) + Float64(5.0 * Float64(Float64(eps * (x ^ 4.0)) + Float64(x * (eps ^ 4.0)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.4e-41], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e-59], N[(N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[((-N[Power[x, 3.0], $MachinePrecision]) * N[(N[Power[eps, 2.0], $MachinePrecision] * -10.0), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * N[(10.0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4000000000000001e-41

   1. Initial program 31.8%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

   if -1.4000000000000001e-41 < x < 3.6999999999999999e-59

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   6. Simplified 100.0%

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

   if 3.6999999999999999e-59 < x

   1. Initial program 39.2%

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

   3. Taylor expanded in x around -inf 93.2%

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

   5. Simplified 93.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-{x}^{3}, {\varepsilon}^{2} \cdot -10, {x}^{2} \cdot \left(10 \cdot {\varepsilon}^{3}\right)\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4} + x \cdot {\varepsilon}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-311} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-311) (not (<= t_0 0.0)))
     t_0
     (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-311) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-1d-311)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    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 <= -1e-311) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	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 <= -1e-311) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -1e-311) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	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 <= -1e-311) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	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[Or[LessEqual[t$95$0, -1e-311], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.99999999999948e-312 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 96.5%

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

   if -9.99999999999948e-312 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-311} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot {x}^{4}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon \cdot t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-56}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (pow x 4.0))))
   (if (<= x -4.5e-42)
     (* eps t_0)
     (if (<= x 2.2e-56)
       (+ (* x (* 5.0 (pow eps 4.0))) (pow eps 5.0))
       (* eps (+ t_0 (* eps (* (pow x 3.0) 10.0))))))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * pow(x, 4.0);
	double tmp;
	if (x <= -4.5e-42) {
		tmp = eps * t_0;
	} else if (x <= 2.2e-56) {
		tmp = (x * (5.0 * pow(eps, 4.0))) + pow(eps, 5.0);
	} else {
		tmp = eps * (t_0 + (eps * (pow(x, 3.0) * 10.0)));
	}
	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 * (x ** 4.0d0)
    if (x <= (-4.5d-42)) then
        tmp = eps * t_0
    else if (x <= 2.2d-56) then
        tmp = (x * (5.0d0 * (eps ** 4.0d0))) + (eps ** 5.0d0)
    else
        tmp = eps * (t_0 + (eps * ((x ** 3.0d0) * 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 5.0 * Math.pow(x, 4.0);
	double tmp;
	if (x <= -4.5e-42) {
		tmp = eps * t_0;
	} else if (x <= 2.2e-56) {
		tmp = (x * (5.0 * Math.pow(eps, 4.0))) + Math.pow(eps, 5.0);
	} else {
		tmp = eps * (t_0 + (eps * (Math.pow(x, 3.0) * 10.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 5.0 * math.pow(x, 4.0)
	tmp = 0
	if x <= -4.5e-42:
		tmp = eps * t_0
	elif x <= 2.2e-56:
		tmp = (x * (5.0 * math.pow(eps, 4.0))) + math.pow(eps, 5.0)
	else:
		tmp = eps * (t_0 + (eps * (math.pow(x, 3.0) * 10.0)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(5.0 * (x ^ 4.0))
	tmp = 0.0
	if (x <= -4.5e-42)
		tmp = Float64(eps * t_0);
	elseif (x <= 2.2e-56)
		tmp = Float64(Float64(x * Float64(5.0 * (eps ^ 4.0))) + (eps ^ 5.0));
	else
		tmp = Float64(eps * Float64(t_0 + Float64(eps * Float64((x ^ 3.0) * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 5.0 * (x ^ 4.0);
	tmp = 0.0;
	if (x <= -4.5e-42)
		tmp = eps * t_0;
	elseif (x <= 2.2e-56)
		tmp = (x * (5.0 * (eps ^ 4.0))) + (eps ^ 5.0);
	else
		tmp = eps * (t_0 + (eps * ((x ^ 3.0) * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e-42], N[(eps * t$95$0), $MachinePrecision], If[LessEqual[x, 2.2e-56], N[(N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(t$95$0 + N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5e-42

   1. Initial program 31.8%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

   if -4.5e-42 < x < 2.20000000000000004e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   6. Simplified 100.0%

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

   if 2.20000000000000004e-56 < x

   1. Initial program 39.2%

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

   3. Taylor expanded in x around inf 93.0%

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

   5. Simplified 93.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-56}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \sqrt{{x}^{8} \cdot 25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.2e-41)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1e-60) (pow eps 5.0) (* eps (sqrt (* (pow x 8.0) 25.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.2e-41) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1e-60) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * sqrt((pow(x, 8.0) * 25.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.2d-41)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 1d-60) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * sqrt(((x ** 8.0d0) * 25.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.2e-41) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 1e-60) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * Math.sqrt((Math.pow(x, 8.0) * 25.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.2e-41:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 1e-60:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * math.sqrt((math.pow(x, 8.0) * 25.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.2e-41)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1e-60)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * sqrt(Float64((x ^ 8.0) * 25.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.2e-41)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 1e-60)
		tmp = eps ^ 5.0;
	else
		tmp = eps * sqrt(((x ^ 8.0) * 25.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.2e-41], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-60], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[Sqrt[N[(N[Power[x, 8.0], $MachinePrecision] * 25.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.20000000000000011e-41

   1. Initial program 31.8%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

   if -1.20000000000000011e-41 < x < 9.9999999999999997e-61

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

   if 9.9999999999999997e-61 < x

   1. Initial program 39.2%

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

   3. Taylor expanded in x around inf 91.0%

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

      1. *-commutative 91.0%

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

      2. distribute-rgt1-in 91.0%

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

      3. metadata-eval 91.0%

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

      4. *-commutative 91.0%

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

      5. associate-*r* 91.1%

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

   5. Simplified 91.1%

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

      1. add-sqr-sqrt 90.8%

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

      2. sqrt-unprod 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

      5. swap-sqr 91.0%

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

      6. pow-prod-up 91.2%

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

      7. metadata-eval 91.2%

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

      8. metadata-eval 91.2%

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

   7. Applied egg-rr 91.2%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \sqrt{{x}^{8} \cdot 25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-41} \lor \neg \left(x \leq 3.2 \cdot 10^{-56}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8e-41) (not (<= x 3.2e-56)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8e-41) || !(x <= 3.2e-56)) {
		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 <= (-8d-41)) .or. (.not. (x <= 3.2d-56))) 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 <= -8e-41) || !(x <= 3.2e-56)) {
		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 <= -8e-41) or not (x <= 3.2e-56):
		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 <= -8e-41) || !(x <= 3.2e-56))
		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 <= -8e-41) || ~((x <= 3.2e-56)))
		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, -8e-41], N[Not[LessEqual[x, 3.2e-56]], $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 < -8.00000000000000005e-41 or 3.19999999999999986e-56 < x

   1. Initial program 36.4%

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

   3. Taylor expanded in x around inf 94.2%

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

      1. *-commutative 94.2%

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

      2. distribute-rgt1-in 94.2%

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

      3. metadata-eval 94.2%

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

      4. *-commutative 94.2%

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

      5. associate-*r* 94.3%

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

   5. Simplified 94.3%

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

   6. Taylor expanded in eps around 0 94.1%

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

   if -8.00000000000000005e-41 < x < 3.19999999999999986e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-41} \lor \neg \left(x \leq 3.2 \cdot 10^{-56}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-41} \lor \neg \left(x \leq 3.3 \cdot 10^{-56}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.25e-41) (not (<= x 3.3e-56)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.25e-41) || !(x <= 3.3e-56)) {
		tmp = eps * (5.0 * 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 <= (-1.25d-41)) .or. (.not. (x <= 3.3d-56))) then
        tmp = eps * (5.0d0 * (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 <= -1.25e-41) || !(x <= 3.3e-56)) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.25e-41) or not (x <= 3.3e-56):
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.25e-41) || !(x <= 3.3e-56))
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.25e-41) || ~((x <= 3.3e-56)))
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.25e-41], N[Not[LessEqual[x, 3.3e-56]], $MachinePrecision]], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2499999999999999e-41 or 3.29999999999999984e-56 < x

   1. Initial program 36.4%

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

   3. Taylor expanded in x around inf 94.2%

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

      1. *-commutative 94.2%

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

      2. distribute-rgt1-in 94.2%

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

      3. metadata-eval 94.2%

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

      4. *-commutative 94.2%

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

      5. associate-*r* 94.3%

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

   5. Simplified 94.3%

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

   if -1.2499999999999999e-41 < x < 3.29999999999999984e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-41} \lor \neg \left(x \leq 3.3 \cdot 10^{-56}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% 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 89.3%

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

3. Taylor expanded in x around 0 88.6%

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

4. Final simplification 88.6%

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

Reproduce

herbie shell --seed 2024026 
(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)))