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

Percentage Accurate: 88.3% → 98.1%

Time: 7.8s

Alternatives: 8

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% 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: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, \varepsilon, \frac{-10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{-x}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, 10 \cdot \varepsilon\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.5e-58)
   (* (* (* x x) (* x x)) (fma 5.0 eps (/ (* -10.0 (* eps eps)) (- x))))
   (if (<= x 1.25e-51)
     (pow eps 5.0)
     (* (* (* (* (fma x 5.0 (* 10.0 eps)) x) eps) x) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.5e-58) {
		tmp = ((x * x) * (x * x)) * fma(5.0, eps, ((-10.0 * (eps * eps)) / -x));
	} else if (x <= 1.25e-51) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (((fma(x, 5.0, (10.0 * eps)) * x) * eps) * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.5e-58)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * fma(5.0, eps, Float64(Float64(-10.0 * Float64(eps * eps)) / Float64(-x))));
	elseif (x <= 1.25e-51)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, Float64(10.0 * eps)) * x) * eps) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.5e-58], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(5.0 * eps + N[(N[(-10.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-51], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(N[(x * 5.0 + N[(10.0 * eps), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999994e-58

   1. Initial program 41.2%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   if -9.4999999999999994e-58 < x < 1.25000000000000001e-51

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.25000000000000001e-51 < x

   1. Initial program 26.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 97.1%

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

   7. Applied rewrites 96.9%

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

   9. Applied rewrites 97.1%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, \varepsilon, \frac{-10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{-x}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, 10 \cdot \varepsilon\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 85.0%

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

7. Applied rewrites 85.0%

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

8. Final simplification 85.0%

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

Alternative 3: 83.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + {x}^{4}\right)\right)} \]
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 85.0%

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

7. Applied rewrites 85.0%

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

9. Applied rewrites 85.0%

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

10. Final simplification 85.0%

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

Alternative 4: 82.9% 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 87.4%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + {x}^{4}\right)\right)} \]
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 85.0%

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

7. Applied rewrites 85.0%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 84.9%

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

11. Final simplification 84.9%

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

Alternative 5: 82.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 84.8

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

5. Applied rewrites 84.8%

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

7. Applied rewrites 84.8%

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

9. Applied rewrites 84.8%

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

10. Final simplification 84.8%

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

Alternative 6: 82.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 84.8

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

5. Applied rewrites 84.8%

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

7. Applied rewrites 84.8%

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

8. Final simplification 84.8%

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

Alternative 7: 82.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 84.8

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

5. Applied rewrites 84.8%

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

7. Applied rewrites 84.8%

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

8. Final simplification 84.8%

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

Alternative 8: 82.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + {x}^{4}\right)\right)} \]
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 85.0%

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

7. Applied rewrites 85.0%

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

9. Applied rewrites 85.0%

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

10. Taylor expanded in eps around 0

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

12. Applied rewrites 84.8%

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

13. Final simplification 84.8%

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

Reproduce

herbie shell --seed 2024255 
(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)))