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

Percentage Accurate: 91.4% → 96.0%

Time: 15.5s

Alternatives: 7

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.4% 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: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-42}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \mathsf{fma}\left({\varepsilon}^{2}, {x}^{3} \cdot 10, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3}\right) + 5 \cdot {\varepsilon}^{4}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.8e-42)
   (- (pow (+ x eps) 5.0) (pow x 5.0))
   (fma
    eps
    (* 5.0 (pow x 4.0))
    (fma
     (pow eps 2.0)
     (* (pow x 3.0) 10.0)
     (* x (+ (* x (* 10.0 (pow eps 3.0))) (* 5.0 (pow eps 4.0))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.8e-42) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = fma(eps, (5.0 * pow(x, 4.0)), fma(pow(eps, 2.0), (pow(x, 3.0) * 10.0), (x * ((x * (10.0 * pow(eps, 3.0))) + (5.0 * pow(eps, 4.0))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.8e-42)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = fma(eps, Float64(5.0 * (x ^ 4.0)), fma((eps ^ 2.0), Float64((x ^ 3.0) * 10.0), Float64(x * Float64(Float64(x * Float64(10.0 * (eps ^ 3.0))) + Float64(5.0 * (eps ^ 4.0))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.8e-42], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] + N[(x * N[(N[(x * N[(10.0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999998e-42

   1. Initial program 96.8%

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

   if 2.79999999999999998e-42 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf 94.9%

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

   5. Simplified 95.2%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-42}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \mathsf{fma}\left({\varepsilon}^{2}, {x}^{3} \cdot 10, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3}\right) + 5 \cdot {\varepsilon}^{4}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \left({\varepsilon}^{2} \cdot 10\right) \cdot \left(\left(x + \varepsilon\right) \cdot {x}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.6e-43)
   (- (pow (+ x eps) 5.0) (pow x 5.0))
   (fma
    eps
    (* 5.0 (pow x 4.0))
    (* (* (pow eps 2.0) 10.0) (* (+ x eps) (pow x 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.6e-43) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = fma(eps, (5.0 * pow(x, 4.0)), ((pow(eps, 2.0) * 10.0) * ((x + eps) * pow(x, 2.0))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.6e-43)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = fma(eps, Float64(5.0 * (x ^ 4.0)), Float64(Float64((eps ^ 2.0) * 10.0) * Float64(Float64(x + eps) * (x ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.6e-43], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * 10.0), $MachinePrecision] * N[(N[(x + eps), $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999992e-43

   1. Initial program 96.8%

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

   if 1.59999999999999992e-43 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf 94.5%

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

   5. Simplified 94.8%

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

      1. expm1-log1p-u 94.8%

         \[\leadsto \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10 + \varepsilon \cdot \left({x}^{2} \cdot 10\right)\right)\right)\right)}\right) \]

      2. expm1-udef 92.0%

         \[\leadsto \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \color{blue}{e^{\mathsf{log1p}\left({\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10 + \varepsilon \cdot \left({x}^{2} \cdot 10\right)\right)\right)} - 1}\right) \]

      3. +-commutative 92.0%

         \[\leadsto \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, e^{\mathsf{log1p}\left({\varepsilon}^{2} \cdot \color{blue}{\left(\varepsilon \cdot \left({x}^{2} \cdot 10\right) + {x}^{3} \cdot 10\right)}\right)} - 1\right) \]

      4. associate-*r* 92.0%

         \[\leadsto \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, e^{\mathsf{log1p}\left({\varepsilon}^{2} \cdot \left(\color{blue}{\left(\varepsilon \cdot {x}^{2}\right) \cdot 10} + {x}^{3} \cdot 10\right)\right)} - 1\right) \]

      5. distribute-rgt-out 92.0%

         \[\leadsto \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, e^{\mathsf{log1p}\left({\varepsilon}^{2} \cdot \color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)\right)}\right)} - 1\right) \]

   7. Applied egg-rr 92.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \color{blue}{e^{\mathsf{log1p}\left({\varepsilon}^{2} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)\right)\right)} - 1}\right) \]
   8. Step-by-step derivation

      1. expm1-def 94.8%

         \[\leadsto \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\varepsilon}^{2} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)\right)\right)\right)}\right) \]

      2. expm1-log1p 94.8%

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

      3. associate-*r* 94.8%

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

      4. *-commutative 94.8%

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

      5. cube-mult 94.8%

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

      6. unpow2 94.8%

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

      7. distribute-rgt-out 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \left({\varepsilon}^{2} \cdot 10\right) \cdot \left(\left(x + \varepsilon\right) \cdot {x}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.5e-43)
   (- (pow (+ x eps) 5.0) (pow x 5.0))
   (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* (pow x 3.0) 10.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.5e-43) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (pow(x, 3.0) * 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.5d-43) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * ((x ** 3.0d0) * 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.5e-43) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (Math.pow(x, 3.0) * 10.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.5e-43:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (math.pow(x, 3.0) * 10.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.5e-43)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64((x ^ 3.0) * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.5e-43)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * ((x ^ 3.0) * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.5e-43], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.50000000000000002e-43

   1. Initial program 96.8%

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

   if 1.50000000000000002e-43 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf 93.0%

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

   5. Simplified 93.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.1e-43)
   (- (pow (+ x eps) 5.0) (pow x 5.0))
   (* eps (* 5.0 (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.1e-43) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.1d-43) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.1e-43) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.1e-43:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.1e-43)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.1e-43)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.1e-43], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.09999999999999999e-43

   1. Initial program 96.8%

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

   if 1.09999999999999999e-43 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

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

      4. *-commutative 91.7%

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

      5. associate-*r* 92.0%

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

   5. Simplified 92.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 8.2e-44) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 8.2e-44) {
		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 <= 8.2d-44) 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 <= 8.2e-44) {
		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 <= 8.2e-44:
		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 <= 8.2e-44)
		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 <= 8.2e-44)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 8.2e-44], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.19999999999999984e-44

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0 94.9%

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

   if 8.19999999999999984e-44 < x

   1. Initial program 44.3%

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

      1. add-cube-cbrt 44.4%

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

      2. pow2 44.4%

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

   4. Applied egg-rr 44.4%

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

   5. Taylor expanded in x around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

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

      4. associate-*l* 91.7%

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

   7. Simplified 91.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 8.5e-44) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 8.5e-44) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 8.5d-44) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 8.5e-44) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 8.5e-44:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 8.5e-44)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8.5e-44)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 8.5e-44], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.5000000000000002e-44

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0 94.9%

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

   if 8.5000000000000002e-44 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

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

      4. *-commutative 91.7%

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

      5. associate-*r* 92.0%

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

   5. Simplified 92.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.4% 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 92.9%

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

3. Taylor expanded in x around 0 90.6%

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

4. Final simplification 90.6%

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

Reproduce

herbie shell --seed 2024031 
(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)))