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

Percentage Accurate: 88.9% → 99.5%

Time: 8.3s

Alternatives: 8

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: 88.9% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-317) (not (<= t_0 0.0)))
     t_0
     (* 5.0 (* eps (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-317) || !(t_0 <= 0.0)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-1d-317)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = 5.0d0 * (eps * (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-317) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = 5.0 * (eps * 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-317) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = 5.0 * (eps * 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-317) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(5.0 * Float64(eps * (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-317) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = 5.0 * (eps * (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-317], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.00000023e-317 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 95.4%

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

   if -1.00000023e-317 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-317} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-41} \lor \neg \left(x \leq 2.4 \cdot 10^{-50}\right):\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.15e-41) (not (<= x 2.4e-50)))
   (* (pow x 3.0) (* eps (+ (* x 5.0) (* eps 10.0))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.15e-41) || !(x <= 2.4e-50)) {
		tmp = pow(x, 3.0) * (eps * ((x * 5.0) + (eps * 10.0)));
	} else {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.15d-41)) .or. (.not. (x <= 2.4d-50))) then
        tmp = (x ** 3.0d0) * (eps * ((x * 5.0d0) + (eps * 10.0d0)))
    else
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.15e-41) || !(x <= 2.4e-50)) {
		tmp = Math.pow(x, 3.0) * (eps * ((x * 5.0) + (eps * 10.0)));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.15e-41) or not (x <= 2.4e-50):
		tmp = math.pow(x, 3.0) * (eps * ((x * 5.0) + (eps * 10.0)))
	else:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.15e-41) || !(x <= 2.4e-50))
		tmp = Float64((x ^ 3.0) * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	else
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.15e-41) || ~((x <= 2.4e-50)))
		tmp = (x ^ 3.0) * (eps * ((x * 5.0) + (eps * 10.0)));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.15e-41], N[Not[LessEqual[x, 2.4e-50]], $MachinePrecision]], N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1499999999999999e-41 or 2.40000000000000002e-50 < x

   1. Initial program 40.1%

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

   3. Taylor expanded in eps around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. associate-+r+ 86.9%

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

      3. distribute-lft1-in 86.9%

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

      4. metadata-eval 86.9%

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

      5. *-commutative 86.9%

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

      6. distribute-rgt-out 86.9%

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

      7. associate-*r* 86.9%

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

      8. unpow2 86.9%

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

      9. cube-mult 86.9%

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

      10. distribute-lft-out 86.9%

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

      11. metadata-eval 86.9%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in x around 0 86.7%

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

   7. Taylor expanded in eps around 0 86.9%

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

   if -2.1499999999999999e-41 < x < 2.40000000000000002e-50

   1. Initial program 99.7%

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

   3. Taylor expanded in eps around inf 99.6%

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

      1. distribute-lft1-in 99.6%

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

      2. metadata-eval 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-41} \lor \neg \left(x \leq 2.4 \cdot 10^{-50}\right):\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-41}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-50}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.6e-41)
   (* (pow x 3.0) (* eps (* x 5.0)))
   (if (<= x 2.05e-50)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.6e-41) {
		tmp = pow(x, 3.0) * (eps * (x * 5.0));
	} else if (x <= 2.05e-50) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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 <= (-4.6d-41)) then
        tmp = (x ** 3.0d0) * (eps * (x * 5.0d0))
    else if (x <= 2.05d-50) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    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 <= -4.6e-41) {
		tmp = Math.pow(x, 3.0) * (eps * (x * 5.0));
	} else if (x <= 2.05e-50) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.6e-41:
		tmp = math.pow(x, 3.0) * (eps * (x * 5.0))
	elif x <= 2.05e-50:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.6e-41)
		tmp = Float64((x ^ 3.0) * Float64(eps * Float64(x * 5.0)));
	elseif (x <= 2.05e-50)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	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 <= -4.6e-41)
		tmp = (x ^ 3.0) * (eps * (x * 5.0));
	elseif (x <= 2.05e-50)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.6e-41], N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e-50], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6000000000000002e-41

   1. Initial program 43.4%

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

   3. Taylor expanded in eps around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. associate-+r+ 87.5%

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

      3. distribute-lft1-in 87.5%

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

      4. metadata-eval 87.5%

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

      5. *-commutative 87.5%

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

      6. distribute-rgt-out 87.5%

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

      7. associate-*r* 87.5%

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

      8. unpow2 87.5%

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

      9. cube-mult 87.5%

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

      10. distribute-lft-out 87.5%

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

      11. metadata-eval 87.5%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in x around 0 87.4%

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

   7. Taylor expanded in eps around 0 87.3%

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

      1. associate-*r* 87.5%

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

      2. *-commutative 87.5%

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

   9. Simplified 87.5%

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

   10. Taylor expanded in x around 0 87.3%

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

       1. associate-*r* 87.5%

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

       2. *-commutative 87.5%

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

       3. associate-*r* 87.5%

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

   12. Simplified 87.5%

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

   if -4.6000000000000002e-41 < x < 2.04999999999999993e-50

   1. Initial program 99.7%

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

   3. Taylor expanded in eps around inf 99.6%

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

      1. distribute-lft1-in 99.6%

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

      2. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

   if 2.04999999999999993e-50 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf 84.7%

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

      1. distribute-rgt1-in 84.7%

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

      2. metadata-eval 84.7%

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

   5. Simplified 84.7%

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

   6. Taylor expanded in x around 0 85.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-41}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-50}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-41} \lor \neg \left(x \leq 5 \cdot 10^{-51}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.5e-41) (not (<= x 5e-51)))
   (* 5.0 (* eps (pow x 4.0)))
   (* (pow eps 4.0) (+ eps (* x 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.5e-41) || !(x <= 5e-51)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 4.0) * (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 <= (-4.5d-41)) .or. (.not. (x <= 5d-51))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -4.5e-41) || !(x <= 5e-51)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -4.5e-41) or not (x <= 5e-51):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -4.5e-41) || !(x <= 5e-51))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -4.5e-41) || ~((x <= 5e-51)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -4.5e-41], N[Not[LessEqual[x, 5e-51]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e-41 or 5.00000000000000004e-51 < x

   1. Initial program 40.1%

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

   3. Taylor expanded in x around inf 85.5%

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

      1. distribute-rgt1-in 85.5%

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

      2. metadata-eval 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0 85.8%

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

   if -4.5e-41 < x < 5.00000000000000004e-51

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. fma-define 99.6%

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

      2. distribute-lft1-in 99.6%

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

      3. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around 0 99.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-41} \lor \neg \left(x \leq 5 \cdot 10^{-51}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-41}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-41)
   (* (pow x 3.0) (* eps (* x 5.0)))
   (if (<= x 1.05e-50)
     (* (pow eps 4.0) (+ eps (* x 5.0)))
     (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-41) {
		tmp = pow(x, 3.0) * (eps * (x * 5.0));
	} else if (x <= 1.05e-50) {
		tmp = pow(eps, 4.0) * (eps + (x * 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 <= (-2d-41)) then
        tmp = (x ** 3.0d0) * (eps * (x * 5.0d0))
    else if (x <= 1.05d-50) then
        tmp = (eps ** 4.0d0) * (eps + (x * 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 <= -2e-41) {
		tmp = Math.pow(x, 3.0) * (eps * (x * 5.0));
	} else if (x <= 1.05e-50) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-41:
		tmp = math.pow(x, 3.0) * (eps * (x * 5.0))
	elif x <= 1.05e-50:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-41)
		tmp = Float64((x ^ 3.0) * Float64(eps * Float64(x * 5.0)));
	elseif (x <= 1.05e-50)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 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 <= -2e-41)
		tmp = (x ^ 3.0) * (eps * (x * 5.0));
	elseif (x <= 1.05e-50)
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-41], N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-50], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000001e-41

   1. Initial program 43.4%

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

   3. Taylor expanded in eps around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. associate-+r+ 87.5%

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

      3. distribute-lft1-in 87.5%

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

      4. metadata-eval 87.5%

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

      5. *-commutative 87.5%

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

      6. distribute-rgt-out 87.5%

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

      7. associate-*r* 87.5%

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

      8. unpow2 87.5%

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

      9. cube-mult 87.5%

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

      10. distribute-lft-out 87.5%

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

      11. metadata-eval 87.5%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot \left(4 + \color{blue}{6}\right)\right)\right) \]

      12. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in x around 0 87.4%

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

   7. Taylor expanded in eps around 0 87.3%

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

      1. associate-*r* 87.5%

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

      2. *-commutative 87.5%

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

   9. Simplified 87.5%

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

   10. Taylor expanded in x around 0 87.3%

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

       1. associate-*r* 87.5%

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

       2. *-commutative 87.5%

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

       3. associate-*r* 87.5%

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

   12. Simplified 87.5%

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

   if -2.00000000000000001e-41 < x < 1.05e-50

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. fma-define 99.6%

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

      2. distribute-lft1-in 99.6%

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

      3. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around 0 99.4%

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

   if 1.05e-50 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf 84.7%

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

      1. distribute-rgt1-in 84.7%

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

      2. metadata-eval 84.7%

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

   5. Simplified 84.7%

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

   6. Taylor expanded in x around 0 85.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-41}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-41} \lor \neg \left(x \leq 2.8 \cdot 10^{-50}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.75e-41) (not (<= x 2.8e-50)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.75e-41) || !(x <= 2.8e-50)) {
		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 <= (-1.75d-41)) .or. (.not. (x <= 2.8d-50))) 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 <= -1.75e-41) || !(x <= 2.8e-50)) {
		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 <= -1.75e-41) or not (x <= 2.8e-50):
		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 <= -1.75e-41) || !(x <= 2.8e-50))
		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 <= -1.75e-41) || ~((x <= 2.8e-50)))
		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, -1.75e-41], N[Not[LessEqual[x, 2.8e-50]], $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 < -1.75e-41 or 2.7999999999999998e-50 < x

   1. Initial program 40.1%

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

   3. Taylor expanded in x around inf 85.5%

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

      1. distribute-rgt1-in 85.5%

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

      2. metadata-eval 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0 85.8%

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

   if -1.75e-41 < x < 2.7999999999999998e-50

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-41} \lor \neg \left(x \leq 2.8 \cdot 10^{-50}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.0% 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 88.3%

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

3. Taylor expanded in x around 0 86.8%

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

4. Final simplification 86.8%

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

Alternative 8: 70.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 88.3%

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

   1. sub-neg 88.3%

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

   2. +-commutative 88.3%

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

   3. add-cube-cbrt 82.6%

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

   4. distribute-lft-neg-in 82.6%

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

   5. fma-define 81.1%

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

   6. cbrt-unprod 79.9%

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

   7. pow-prod-up 79.9%

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

   8. metadata-eval 79.9%

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

4. Applied egg-rr 79.9%

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

5. Taylor expanded in eps around 0 69.9%

   \[\leadsto \color{blue}{-1 \cdot {x}^{5} + {x}^{5}} \]
6. Step-by-step derivation

   1. distribute-lft1-in 69.9%

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

   2. metadata-eval 69.9%

      \[\leadsto \color{blue}{0} \cdot {x}^{5} \]

   3. mul0-lft 69.9%

      \[\leadsto \color{blue}{0} \]

7. Simplified 69.9%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 69.9%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))