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

Percentage Accurate: 88.5% → 97.9%

Time: 9.3s

Alternatives: 6

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: 88.5% 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: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-47} \lor \neg \left(x \leq 4.5 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(-{x}^{3}, {\varepsilon}^{2} \cdot -10, {x}^{2} \cdot \left(10 \cdot {\varepsilon}^{3}\right)\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4} + x \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.5e-47) (not (<= x 4.5e-41)))
   (+
    (fma
     (- (pow x 3.0))
     (* (pow eps 2.0) -10.0)
     (* (pow x 2.0) (* 10.0 (pow eps 3.0))))
    (* 5.0 (+ (* eps (pow x 4.0)) (* x (pow eps 4.0)))))
   (+ (* x (* 5.0 (pow eps 4.0))) (pow eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.5e-47) || !(x <= 4.5e-41)) {
		tmp = fma(-pow(x, 3.0), (pow(eps, 2.0) * -10.0), (pow(x, 2.0) * (10.0 * pow(eps, 3.0)))) + (5.0 * ((eps * pow(x, 4.0)) + (x * pow(eps, 4.0))));
	} else {
		tmp = (x * (5.0 * pow(eps, 4.0))) + pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.5e-47) || !(x <= 4.5e-41))
		tmp = Float64(fma(Float64(-(x ^ 3.0)), Float64((eps ^ 2.0) * -10.0), Float64((x ^ 2.0) * Float64(10.0 * (eps ^ 3.0)))) + Float64(5.0 * Float64(Float64(eps * (x ^ 4.0)) + Float64(x * (eps ^ 4.0)))));
	else
		tmp = Float64(Float64(x * Float64(5.0 * (eps ^ 4.0))) + (eps ^ 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.5e-47], N[Not[LessEqual[x, 4.5e-41]], $MachinePrecision]], N[(N[((-N[Power[x, 3.0], $MachinePrecision]) * N[(N[Power[eps, 2.0], $MachinePrecision] * -10.0), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * N[(10.0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999998e-47 or 4.5e-41 < x

   1. Initial program 31.7%

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

   3. Taylor expanded in x around -inf 96.7%

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

   5. Simplified 96.8%

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

   if -3.4999999999999998e-47 < x < 4.5e-41

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-47} \lor \neg \left(x \leq 4.5 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(-{x}^{3}, {\varepsilon}^{2} \cdot -10, {x}^{2} \cdot \left(10 \cdot {\varepsilon}^{3}\right)\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4} + x \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot {x}^{4}\\ t_1 := 5 \cdot {\varepsilon}^{4}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot t\_0 - {x}^{3} \cdot \left({\varepsilon}^{2} \cdot -10\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;x \cdot t\_1 + {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t\_0, \mathsf{fma}\left({\varepsilon}^{2}, {x}^{3} \cdot 10, x \cdot \left(t\_1 + x \cdot \left(10 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (pow x 4.0))) (t_1 (* 5.0 (pow eps 4.0))))
   (if (<= x -2.6e-47)
     (- (* eps t_0) (* (pow x 3.0) (* (pow eps 2.0) -10.0)))
     (if (<= x 3.8e-41)
       (+ (* x t_1) (pow eps 5.0))
       (fma
        eps
        t_0
        (fma
         (pow eps 2.0)
         (* (pow x 3.0) 10.0)
         (* x (+ t_1 (* x (* 10.0 (pow eps 3.0)))))))))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * pow(x, 4.0);
	double t_1 = 5.0 * pow(eps, 4.0);
	double tmp;
	if (x <= -2.6e-47) {
		tmp = (eps * t_0) - (pow(x, 3.0) * (pow(eps, 2.0) * -10.0));
	} else if (x <= 3.8e-41) {
		tmp = (x * t_1) + pow(eps, 5.0);
	} else {
		tmp = fma(eps, t_0, fma(pow(eps, 2.0), (pow(x, 3.0) * 10.0), (x * (t_1 + (x * (10.0 * pow(eps, 3.0)))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(5.0 * (x ^ 4.0))
	t_1 = Float64(5.0 * (eps ^ 4.0))
	tmp = 0.0
	if (x <= -2.6e-47)
		tmp = Float64(Float64(eps * t_0) - Float64((x ^ 3.0) * Float64((eps ^ 2.0) * -10.0)));
	elseif (x <= 3.8e-41)
		tmp = Float64(Float64(x * t_1) + (eps ^ 5.0));
	else
		tmp = fma(eps, t_0, fma((eps ^ 2.0), Float64((x ^ 3.0) * 10.0), Float64(x * Float64(t_1 + Float64(x * Float64(10.0 * (eps ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e-47], N[(N[(eps * t$95$0), $MachinePrecision] - N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[Power[eps, 2.0], $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e-41], N[(N[(x * t$95$1), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * t$95$0 + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] + N[(x * N[(t$95$1 + N[(x * N[(10.0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e-47

   1. Initial program 25.4%

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

   3. Taylor expanded in x around -inf 95.2%

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

      1. +-commutative 95.2%

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

      2. mul-1-neg 95.2%

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

      3. unsub-neg 95.2%

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

      4. *-commutative 95.2%

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

      5. distribute-rgt1-in 95.2%

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

      6. metadata-eval 95.2%

         \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} - {x}^{3} \cdot \left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right) \]

      7. *-commutative 95.2%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} - {x}^{3} \cdot \left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right) \]

      8. associate-*r* 95.2%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} - {x}^{3} \cdot \left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right) \]

      9. *-commutative 95.2%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) - {x}^{3} \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot -4} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right) \]

      10. mul-1-neg 95.2%

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

      11. distribute-rgt-out 95.2%

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

      12. distribute-rgt-neg-in 95.2%

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

      13. distribute-lft-out 95.2%

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

   5. Simplified 95.2%

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

   if -2.6e-47 < x < 3.79999999999999979e-41

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 3.79999999999999979e-41 < x

   1. Initial program 43.7%

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

   3. Taylor expanded in x around inf 99.5%

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

   5. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right) - {x}^{3} \cdot \left({\varepsilon}^{2} \cdot -10\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \mathsf{fma}\left({\varepsilon}^{2}, {x}^{3} \cdot 10, x \cdot \left(5 \cdot {\varepsilon}^{4} + x \cdot \left(10 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-323) (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-323) || !(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-323)) .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-323) || !(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-323) 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-323) || !(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-323) || ~((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-323], 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) 5) (pow.f64 x 5)) < -9.88131e-324 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.4%

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

   if -9.88131e-324 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 88.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-323} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-47} \lor \neg \left(x \leq 3.8 \cdot 10^{-41}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.2e-47) (not (<= x 3.8e-41)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.2e-47) || !(x <= 3.8e-41)) {
		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 <= (-1.2d-47)) .or. (.not. (x <= 3.8d-41))) 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 <= -1.2e-47) || !(x <= 3.8e-41)) {
		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 <= -1.2e-47) or not (x <= 3.8e-41):
		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 <= -1.2e-47) || !(x <= 3.8e-41))
		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 <= -1.2e-47) || ~((x <= 3.8e-41)))
		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, -1.2e-47], N[Not[LessEqual[x, 3.8e-41]], $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 < -1.2e-47 or 3.79999999999999979e-41 < x

   1. Initial program 31.7%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. distribute-rgt1-in 93.7%

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

      2. metadata-eval 93.7%

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

   5. Simplified 93.7%

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

   6. Taylor expanded in x around 0 93.8%

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

   if -1.2e-47 < x < 3.79999999999999979e-41

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-47} \lor \neg \left(x \leq 3.8 \cdot 10^{-41}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.4% 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 90.6%

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

3. Taylor expanded in x around 0 89.6%

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

4. Final simplification 89.6%

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

Alternative 6: 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 90.6%

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

   1. sub-neg 90.6%

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

   2. +-commutative 90.6%

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

   3. add-sqr-sqrt 86.1%

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

   4. distribute-rgt-neg-in 86.1%

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

   5. fma-def 83.9%

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

   6. sqrt-pow1 43.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{5}{2}\right)}}, -\sqrt{{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

   7. metadata-eval 43.7%

      \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{2.5}}, -\sqrt{{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

   8. sqrt-pow1 43.7%

      \[\leadsto \mathsf{fma}\left({x}^{2.5}, -\color{blue}{{x}^{\left(\frac{5}{2}\right)}}, {\left(x + \varepsilon\right)}^{5}\right) \]

   9. metadata-eval 43.7%

      \[\leadsto \mathsf{fma}\left({x}^{2.5}, -{x}^{\color{blue}{2.5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

4. Applied egg-rr 43.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2.5}, -{x}^{2.5}, {\left(x + \varepsilon\right)}^{5}\right)} \]

5. Taylor expanded in eps around 0 72.8%

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

   1. distribute-lft1-in 72.8%

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

   2. metadata-eval 72.8%

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

   3. mul0-lft 72.8%

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

7. Simplified 72.8%

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

8. Final simplification 72.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))