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

Percentage Accurate: 88.4% → 98.9%

Time: 6.6s

Alternatives: 10

Speedup: 0.5×

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: 88.4% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right)}{\varepsilon}, x, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-305)
     (* (fma (/ (fma 10.0 (/ x eps) 5.0) eps) x 1.0) (pow eps 5.0))
     (if (<= t_0 0.0) (* (* (pow x 4.0) eps) 5.0) t_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 <= -5e-305) {
		tmp = fma((fma(10.0, (x / eps), 5.0) / eps), x, 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	} else {
		tmp = t_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 <= -5e-305)
		tmp = Float64(fma(Float64(fma(10.0, Float64(x / eps), 5.0) / eps), x, 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = t_0;
	end
	return 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[LessEqual[t$95$0, -5e-305], N[(N[(N[(N[(10.0 * N[(x / eps), $MachinePrecision] + 5.0), $MachinePrecision] / eps), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99999999999999985e-305

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + x \cdot \left(10 \cdot \frac{x}{{\varepsilon}^{2}} + 5 \cdot \frac{1}{\varepsilon}\right)\right) \cdot {\color{blue}{\varepsilon}}^{5} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right)}{\varepsilon}, x, 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right)}{\varepsilon}, x, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-305) (not (<= t_0 0.0)))
     (* (fma (/ (fma 10.0 (/ x eps) 5.0) eps) x 1.0) (pow eps 5.0))
     (* (* (pow x 4.0) eps) 5.0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0)) {
		tmp = fma((fma(10.0, (x / eps), 5.0) / eps), x, 1.0) * pow(eps, 5.0);
	} else {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0))
		tmp = Float64(fma(Float64(fma(10.0, Float64(x / eps), 5.0) / eps), x, 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	end
	return 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, -5e-305], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(10.0 * N[(x / eps), $MachinePrecision] + 5.0), $MachinePrecision] / eps), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $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))) < -4.99999999999999985e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + x \cdot \left(10 \cdot \frac{x}{{\varepsilon}^{2}} + 5 \cdot \frac{1}{\varepsilon}\right)\right) \cdot {\color{blue}{\varepsilon}}^{5} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right)}{\varepsilon}, x, 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-305} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right)}{\varepsilon}, x, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-305)
     (* (* (* (fma (fma x 5.0 eps) eps (* (* x x) 10.0)) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (pow x 4.0) eps) 5.0)
       (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((x * x) * 10.0)) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	} else {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-305)
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	end
	return 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[LessEqual[t$95$0, -5e-305], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99999999999999985e-305

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.3%

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

   10. Applied rewrites 99.4%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, 10 \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 99.5%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 97.3%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 95.3

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

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-305) (not (<= t_0 0.0)))
     (* (* (* (fma (fma x 5.0 eps) eps (* (* x x) 10.0)) eps) eps) eps)
     (* (* (pow x 4.0) eps) 5.0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0)) {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((x * x) * 10.0)) * eps) * eps) * eps;
	} else {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * eps);
	else
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	end
	return 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, -5e-305], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $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))) < -4.99999999999999985e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, 10 \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 97.7%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-305} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-305) (not (<= t_0 0.0)))
     (* (* (* (fma (fma x 5.0 eps) eps (* (* x x) 10.0)) eps) eps) eps)
     (* (* (pow x 4.0) 5.0) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0)) {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((x * x) * 10.0)) * eps) * eps) * eps;
	} else {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * eps);
	else
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	end
	return 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, -5e-305], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $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))) < -4.99999999999999985e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, 10 \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 97.7%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-305} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 5\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-305) (not (<= t_0 0.0)))
     (* (* (* (fma (fma x 5.0 eps) eps (* (* x x) 10.0)) eps) eps) eps)
     (* (* (* x 5.0) x) (* (* eps x) x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0)) {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((x * x) * 10.0)) * eps) * eps) * eps;
	} else {
		tmp = ((x * 5.0) * x) * ((eps * x) * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(x * 5.0) * x) * Float64(Float64(eps * x) * x));
	end
	return 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, -5e-305], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(x * 5.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * x), $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))) < -4.99999999999999985e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, 10 \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
   11. Step-by-step derivation

   12. Applied rewrites 97.7%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-305} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 5\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 5\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-305) (not (<= t_0 0.0)))
     (* (* (* (fma 5.0 x eps) eps) eps) (* eps eps))
     (* (* (* x 5.0) x) (* (* eps x) x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0)) {
		tmp = ((fma(5.0, x, eps) * eps) * eps) * (eps * eps);
	} else {
		tmp = ((x * 5.0) * x) * ((eps * x) * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-305) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(fma(5.0, x, eps) * eps) * eps) * Float64(eps * eps));
	else
		tmp = Float64(Float64(Float64(x * 5.0) * x) * Float64(Float64(eps * x) * x));
	end
	return 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, -5e-305], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 5.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * x), $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))) < -4.99999999999999985e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.9%

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

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-305} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 5\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft1-in N/A

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

   7. lower-*.f64 N/A

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

   8. distribute-lft1-in N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 82.8

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

5. Applied rewrites 82.8%

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

7. Applied rewrites 82.7%

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

9. Applied rewrites 82.8%

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

11. Applied rewrites 82.8%

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

Alternative 9: 81.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft1-in N/A

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

   7. lower-*.f64 N/A

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

   8. distribute-lft1-in N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 82.8

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

5. Applied rewrites 82.8%

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

7. Applied rewrites 82.7%

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

9. Applied rewrites 82.8%

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

11. Applied rewrites 82.8%

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

Alternative 10: 70.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* (* (* (* x x) 10.0) eps) eps) eps))

C

double code(double x, double eps) {
	return ((((x * x) * 10.0) * eps) * eps) * eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((((x * x) * 10.0d0) * eps) * eps) * eps
end function

Java

public static double code(double x, double eps) {
	return ((((x * x) * 10.0) * eps) * eps) * eps;
}

Python

def code(x, eps):
	return ((((x * x) * 10.0) * eps) * eps) * eps

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(Float64(x * x) * 10.0) * eps) * eps) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = ((((x * x) * 10.0) * eps) * eps) * eps;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in eps around -inf

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 82.1%

   \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 89.2%

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

10. Applied rewrites 89.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, 10 \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

11. Taylor expanded in x around inf

    \[\leadsto \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]
12. Step-by-step derivation

13. Applied rewrites 72.3%

    \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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)))