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

Percentage Accurate: 88.7% → 99.4%

Time: 10.6s

Alternatives: 14

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: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_0 - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -5e-311)
     (- t_0 (* (* (* x x) x) (* x x)))
     (if (<= t_1 0.0)
       (* (* (* (* (fma 10.0 eps (* 5.0 x)) x) eps) x) x)
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-311) {
		tmp = t_0 - (((x * x) * x) * (x * x));
	} else if (t_1 <= 0.0) {
		tmp = (((fma(10.0, eps, (5.0 * x)) * x) * eps) * x) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -5e-311)
		tmp = Float64(t_0 - Float64(Float64(Float64(x * x) * x) * Float64(x * x)));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps) * x) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-311], N[(t$95$0 - N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

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))) < -5.00000000000023e-311

   1. Initial program 96.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow3 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

   if -5.00000000000023e-311 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.0%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 99.9%

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

   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. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* (* x x) x) (* x x)))))
   (if (<= t_1 -5e-311)
     t_2
     (if (<= t_1 0.0)
       (* (* (* (* (fma 10.0 eps (* 5.0 x)) x) eps) x) x)
       t_2))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = t_0 - (((x * x) * x) * (x * x));
	double tmp;
	if (t_1 <= -5e-311) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (((fma(10.0, eps, (5.0 * x)) * x) * eps) * x) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(Float64(x * x) * x) * Float64(x * x)))
	tmp = 0.0
	if (t_1 <= -5e-311)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps) * x) * x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-311], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]]

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))) < -5.00000000000023e-311 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow3 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

   if -5.00000000000023e-311 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.0%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(x, \frac{5}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \left(x \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-52)
   (* (* (* (fma (* 5.0 x) x (* (* 10.0 (+ eps x)) eps)) eps) x) x)
   (if (<= x 2e-63)
     (* (pow eps 5.0) (fma x (/ 5.0 eps) 1.0))
     (*
      (fma
       (* (fma 10.0 eps (* 5.0 x)) (* x x))
       x
       (* (* eps eps) (* (fma 10.0 x (* 5.0 eps)) x)))
      eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-52) {
		tmp = ((fma((5.0 * x), x, ((10.0 * (eps + x)) * eps)) * eps) * x) * x;
	} else if (x <= 2e-63) {
		tmp = pow(eps, 5.0) * fma(x, (5.0 / eps), 1.0);
	} else {
		tmp = fma((fma(10.0, eps, (5.0 * x)) * (x * x)), x, ((eps * eps) * (fma(10.0, x, (5.0 * eps)) * x))) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-52)
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * Float64(eps + x)) * eps)) * eps) * x) * x);
	elseif (x <= 2e-63)
		tmp = Float64((eps ^ 5.0) * fma(x, Float64(5.0 / eps), 1.0));
	else
		tmp = Float64(fma(Float64(fma(10.0, eps, Float64(5.0 * x)) * Float64(x * x)), x, Float64(Float64(eps * eps) * Float64(fma(10.0, x, Float64(5.0 * eps)) * x))) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-52], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * N[(eps + x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2e-63], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(x * N[(5.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0999999999999999e-52

   1. Initial program 44.6%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 93.8%

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

   if -3.0999999999999999e-52 < x < 2.00000000000000013e-63

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if 2.00000000000000013e-63 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 97.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(x, \frac{5}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \left(x \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \left(x \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-52)
   (* (* (* (fma (* 5.0 x) x (* (* 10.0 (+ eps x)) eps)) eps) x) x)
   (if (<= x 2e-63)
     (* (pow eps 4.0) (fma x 5.0 eps))
     (*
      (fma
       (* (fma 10.0 eps (* 5.0 x)) (* x x))
       x
       (* (* eps eps) (* (fma 10.0 x (* 5.0 eps)) x)))
      eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-52) {
		tmp = ((fma((5.0 * x), x, ((10.0 * (eps + x)) * eps)) * eps) * x) * x;
	} else if (x <= 2e-63) {
		tmp = pow(eps, 4.0) * fma(x, 5.0, eps);
	} else {
		tmp = fma((fma(10.0, eps, (5.0 * x)) * (x * x)), x, ((eps * eps) * (fma(10.0, x, (5.0 * eps)) * x))) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-52)
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * Float64(eps + x)) * eps)) * eps) * x) * x);
	elseif (x <= 2e-63)
		tmp = Float64((eps ^ 4.0) * fma(x, 5.0, eps));
	else
		tmp = Float64(fma(Float64(fma(10.0, eps, Float64(5.0 * x)) * Float64(x * x)), x, Float64(Float64(eps * eps) * Float64(fma(10.0, x, Float64(5.0 * eps)) * x))) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-52], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * N[(eps + x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2e-63], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0999999999999999e-52

   1. Initial program 44.6%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 93.8%

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

   if -3.0999999999999999e-52 < x < 2.00000000000000013e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.00000000000000013e-63 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 97.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \left(x \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \left(x \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-52)
   (* (* (* (fma (* 5.0 x) x (* (* 10.0 (+ eps x)) eps)) eps) x) x)
   (if (<= x 2e-63)
     (pow eps 5.0)
     (*
      (fma
       (* (fma 10.0 eps (* 5.0 x)) (* x x))
       x
       (* (* eps eps) (* (fma 10.0 x (* 5.0 eps)) x)))
      eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-52) {
		tmp = ((fma((5.0 * x), x, ((10.0 * (eps + x)) * eps)) * eps) * x) * x;
	} else if (x <= 2e-63) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = fma((fma(10.0, eps, (5.0 * x)) * (x * x)), x, ((eps * eps) * (fma(10.0, x, (5.0 * eps)) * x))) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-52)
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * Float64(eps + x)) * eps)) * eps) * x) * x);
	elseif (x <= 2e-63)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(fma(Float64(fma(10.0, eps, Float64(5.0 * x)) * Float64(x * x)), x, Float64(Float64(eps * eps) * Float64(fma(10.0, x, Float64(5.0 * eps)) * x))) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-52], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * N[(eps + x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2e-63], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0999999999999999e-52

   1. Initial program 44.6%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 93.8%

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

   if -3.0999999999999999e-52 < x < 2.00000000000000013e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2.00000000000000013e-63 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 97.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \left(x \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 10, \left(5 \cdot x\right) \cdot \varepsilon\right), \varepsilon, 10 \cdot t\_0\right), \varepsilon, \left(t\_0 \cdot x\right) \cdot 5\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (fma
     (fma (fma (* x x) 10.0 (* (* 5.0 x) eps)) eps (* 10.0 t_0))
     eps
     (* (* t_0 x) 5.0))
    eps)))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	return fma(fma(fma((x * x), 10.0, ((5.0 * x) * eps)), eps, (10.0 * t_0)), eps, ((t_0 * x) * 5.0)) * eps;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(fma(fma(fma(Float64(x * x), 10.0, Float64(Float64(5.0 * x) * eps)), eps, Float64(10.0 * t_0)), eps, Float64(Float64(t_0 * x) * 5.0)) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 10.0 + N[(N[(5.0 * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps + N[(10.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(t$95$0 * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 86.2%

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

7. Applied rewrites 86.1%

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

8. Final simplification 86.1%

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

Alternative 7: 83.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot \varepsilon, \varepsilon, \left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (fma
    (* (fma 10.0 x (* 5.0 eps)) eps)
    eps
    (* (* (fma 10.0 eps (* 5.0 x)) x) x))
   x)
  eps))

C

double code(double x, double eps) {
	return (fma((fma(10.0, x, (5.0 * eps)) * eps), eps, ((fma(10.0, eps, (5.0 * x)) * x) * x)) * x) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(fma(10.0, x, Float64(5.0 * eps)) * eps), eps, Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * x)) * x) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 86.2%

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

7. Applied rewrites 86.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(x \cdot \left(5 \cdot {\varepsilon}^{3} + x \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 86.1%

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

11. Final simplification 86.1%

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

Alternative 8: 83.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* (fma (* 5.0 x) x (* (* 10.0 (+ eps x)) eps)) eps) x) x))

C

double code(double x, double eps) {
	return ((fma((5.0 * x), x, ((10.0 * (eps + x)) * eps)) * eps) * x) * x;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * Float64(eps + x)) * eps)) * eps) * x) * x)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * N[(eps + x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 86.2%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 86.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 86.1%

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

12. Final simplification 86.1%

    \[\leadsto \left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x \]
13. Add Preprocessing

Alternative 9: 83.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* (* (fma 10.0 eps (* 5.0 x)) x) eps) x) x))

C

double code(double x, double eps) {
	return (((fma(10.0, eps, (5.0 * x)) * x) * eps) * x) * x;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps) * x) * x)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 86.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 86.1%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 86.0%

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

12. Final simplification 86.0%

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

Alternative 10: 83.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* (* (fma 10.0 eps (* 5.0 x)) eps) x) x) x))

C

double code(double x, double eps) {
	return (((fma(10.0, eps, (5.0 * x)) * eps) * x) * x) * x;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * eps) * x) * x) * x)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 86.2%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 85.9%

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

9. Final simplification 85.9%

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

Alternative 11: 83.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 86.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 85.7%

   \[\leadsto \left(\left(\left(x \cdot 5\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 85.7%

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

11. Final simplification 85.7%

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

Alternative 12: 83.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 86.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 85.7%

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

9. Final simplification 85.7%

   \[\leadsto \left(\left(\left(5 \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]
10. Add Preprocessing

Alternative 13: 83.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 86.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 86.1%

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

10. Applied rewrites 86.1%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 85.6%

    \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot 5\right) \cdot x \]
14. Add Preprocessing

Alternative 14: 72.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 86.2%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 86.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 71.0%

    \[\leadsto \left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot 10\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

    \[\leadsto 10 \cdot \color{blue}{\left({\varepsilon}^{3} \cdot {x}^{2}\right)} \]
13. Step-by-step derivation

14. Applied rewrites 71.0%

    \[\leadsto \left(\left(\left(\left(10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot \color{blue}{x} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))