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

Percentage Accurate: 87.9% → 99.3%

Time: 6.9s

Alternatives: 8

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.9% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-313} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} + 4 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -2e-313) (not (<= t_0 0.0)))
     t_0
     (* eps (+ (pow x 4.0) (* 4.0 (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 <= -2e-313) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (pow(x, 4.0) + (4.0 * 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 <= (-2d-313)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * ((x ** 4.0d0) + (4.0d0 * (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 <= -2e-313) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (Math.pow(x, 4.0) + (4.0 * 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 <= -2e-313) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * (math.pow(x, 4.0) + (4.0 * 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 <= -2e-313) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64((x ^ 4.0) + Float64(4.0 * (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 <= -2e-313) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * ((x ^ 4.0) + (4.0 * (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, -2e-313], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] + N[(4.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -1.99999999998e-313 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.9%

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

   if -1.99999999998e-313 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 84.0%

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

   2. Taylor expanded in eps around 0 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-313} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} + 4 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-313} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -2e-313) (not (<= t_0 0.0)))
     t_0
     (* eps (* (* x 5.0) (pow x 3.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -2e-313) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	}
	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 <= (-2d-313)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    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 <= -2e-313) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	}
	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 <= -2e-313) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -2e-313) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	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 <= -2e-313) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	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, -2e-313], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -1.99999999998e-313 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.9%

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

   if -1.99999999998e-313 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 84.0%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. add-sqr-sqrt 99.8%

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

      4. pow2 99.8%

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

      5. *-commutative 99.8%

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

      6. sqrt-prod 99.8%

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

      7. sqrt-pow1 99.8%

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

      8. metadata-eval 99.8%

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

      9. pow2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

      2. swap-sqr 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. associate-*l* 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow3 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-313} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \]

Alternative 3: 97.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.5e-39)
   (* eps (* (* x x) (* 5.0 (* x x))))
   (if (<= x 2.65e-75) (pow eps 5.0) (* eps (* (* x 5.0) (pow x 3.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-8.5d-39)) then
        tmp = eps * ((x * x) * (5.0d0 * (x * x)))
    else if (x <= 2.65d-75) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -8.5e-39:
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	elif x <= 2.65e-75:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.5e-39)
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -8.5e-39)
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.5e-39], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-75], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000005e-39

   1. Initial program 12.5%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqr-pow 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow2 99.6%

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

      8. associate-*r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   if -8.5000000000000005e-39 < x < 2.6499999999999999e-75

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 2.6499999999999999e-75 < x

   1. Initial program 41.6%

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

   2. Taylor expanded in eps around 0 89.2%

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

      1. distribute-lft1-in 89.2%

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

      2. metadata-eval 89.2%

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

      3. add-sqr-sqrt 88.9%

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

      4. pow2 88.9%

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

      5. *-commutative 88.9%

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

      6. sqrt-prod 88.7%

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

      7. sqrt-pow1 88.8%

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

      8. metadata-eval 88.8%

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

      9. pow2 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. unpow2 88.8%

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

      2. swap-sqr 88.6%

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

      3. add-sqr-sqrt 89.1%

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

      4. associate-*r* 89.0%

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

      5. *-commutative 89.0%

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

      6. *-commutative 89.0%

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

      7. associate-*r* 89.0%

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

      8. associate-*l* 89.0%

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

      9. associate-*l* 89.0%

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

      10. pow3 89.1%

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

   6. Applied egg-rr 89.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \]

Alternative 4: 97.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.5e-39)
   (* eps (* (* x x) (* 5.0 (* x x))))
   (if (<= x 2.65e-75) (pow eps 5.0) (* (pow x 3.0) (* eps (* x 5.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 3.0) * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-8.5d-39)) then
        tmp = eps * ((x * x) * (5.0d0 * (x * x)))
    else if (x <= 2.65d-75) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 3.0d0) * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 3.0) * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -8.5e-39:
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	elif x <= 2.65e-75:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 3.0) * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.5e-39)
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 3.0) * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -8.5e-39)
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 3.0) * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.5e-39], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-75], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000005e-39

   1. Initial program 12.5%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqr-pow 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow2 99.6%

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

      8. associate-*r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   if -8.5000000000000005e-39 < x < 2.6499999999999999e-75

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 2.6499999999999999e-75 < x

   1. Initial program 41.6%

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

   2. Taylor expanded in eps around 0 89.2%

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

      1. distribute-lft1-in 89.2%

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

      2. metadata-eval 89.2%

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

      3. sqr-pow 89.1%

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

      4. metadata-eval 89.1%

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

      5. pow2 89.1%

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

      6. metadata-eval 89.1%

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

      7. pow2 89.1%

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

      8. associate-*r* 89.0%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in x around 0 89.0%

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

      1. unpow2 89.0%

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

      2. associate-*r* 89.0%

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

      3. *-commutative 89.0%

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

   7. Simplified 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*r* 89.0%

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

      3. cube-mult 89.1%

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

      4. add-cube-cbrt 88.2%

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

      5. pow3 88.2%

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

      6. associate-*r* 88.2%

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

      7. cbrt-prod 87.8%

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

      8. *-commutative 87.8%

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

      9. rem-cbrt-cube 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. *-commutative 88.0%

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

       2. cube-prod 88.2%

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

       3. rem-cube-cbrt 89.2%

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

   11. Simplified 89.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.5e-39)
   (* eps (* (* x x) (* 5.0 (* x x))))
   (if (<= x 2.65e-75) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-8.5d-39)) then
        tmp = eps * ((x * x) * (5.0d0 * (x * x)))
    else if (x <= 2.65d-75) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -8.5e-39:
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	elif x <= 2.65e-75:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.5e-39)
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -8.5e-39)
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.5e-39], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-75], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000005e-39

   1. Initial program 12.5%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqr-pow 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow2 99.6%

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

      8. associate-*r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   if -8.5000000000000005e-39 < x < 2.6499999999999999e-75

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 2.6499999999999999e-75 < x

   1. Initial program 41.6%

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

   2. Taylor expanded in x around inf 89.1%

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

      1. distribute-lft1-in 89.1%

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

      2. metadata-eval 89.1%

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

      3. associate-*l* 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 6: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.5e-39)
   (* eps (* (* x x) (* 5.0 (* x x))))
   (if (<= x 2.65e-75) (pow eps 5.0) (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-8.5d-39)) then
        tmp = eps * ((x * x) * (5.0d0 * (x * x)))
    else if (x <= 2.65d-75) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -8.5e-39) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else if (x <= 2.65e-75) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -8.5e-39:
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	elif x <= 2.65e-75:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.5e-39)
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -8.5e-39)
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	elseif (x <= 2.65e-75)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.5e-39], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-75], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000005e-39

   1. Initial program 12.5%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqr-pow 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow2 99.6%

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

      8. associate-*r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   if -8.5000000000000005e-39 < x < 2.6499999999999999e-75

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 2.6499999999999999e-75 < x

   1. Initial program 41.6%

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

   2. Taylor expanded in x around inf 89.1%

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

      1. distribute-lft1-in 89.1%

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

      2. metadata-eval 89.1%

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

      3. *-commutative 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-75}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternative 7: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39} \lor \neg \left(x \leq 2.65 \cdot 10^{-75}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8.5e-39) (not (<= x 2.65e-75)))
   (* eps (* (* x x) (* 5.0 (* x x))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8.5e-39) || !(x <= 2.65e-75)) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-8.5d-39)) .or. (.not. (x <= 2.65d-75))) then
        tmp = eps * ((x * x) * (5.0d0 * (x * x)))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -8.5e-39) || !(x <= 2.65e-75)) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -8.5e-39) or not (x <= 2.65e-75):
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -8.5e-39) || !(x <= 2.65e-75))
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -8.5e-39) || ~((x <= 2.65e-75)))
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -8.5e-39], N[Not[LessEqual[x, 2.65e-75]], $MachinePrecision]], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000005e-39 or 2.6499999999999999e-75 < x

   1. Initial program 34.9%

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

   2. Taylor expanded in eps around 0 91.6%

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

      1. distribute-lft1-in 91.6%

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

      2. metadata-eval 91.6%

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

      3. sqr-pow 91.5%

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

      4. metadata-eval 91.5%

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

      5. pow2 91.5%

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

      6. metadata-eval 91.5%

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

      7. pow2 91.5%

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

      8. associate-*r* 91.5%

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

   4. Applied egg-rr 91.5%

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

   if -8.5000000000000005e-39 < x < 2.6499999999999999e-75

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-39} \lor \neg \left(x \leq 2.65 \cdot 10^{-75}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 8: 82.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

2. Taylor expanded in eps around 0 82.6%

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

   1. distribute-lft1-in 82.6%

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

   2. metadata-eval 82.6%

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

   3. sqr-pow 82.5%

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

   4. metadata-eval 82.5%

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

   5. pow2 82.5%

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

   6. metadata-eval 82.5%

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

   7. pow2 82.5%

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

   8. associate-*r* 82.5%

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

4. Applied egg-rr 82.5%

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

5. Final simplification 82.5%

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

Reproduce

herbie shell --seed 2023178 
(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)))