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

Percentage Accurate: 88.9% → 97.7%

Time: 11.5s

Alternatives: 17

Speedup: 4.7×

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: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -5e-250)
     (- t_0 (* (* x x) (* x (* x x))))
     (if (<= t_1 0.0) (* eps (* 5.0 (pow x 4.0))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = t_0 - ((x * x) * (x * (x * x)));
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    if (t_1 <= (-5d-250)) then
        tmp = t_0 - ((x * x) * (x * (x * x)))
    else if (t_1 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = t_0 - ((x * x) * (x * (x * x)));
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	tmp = 0
	if t_1 <= -5e-250:
		tmp = t_0 - ((x * x) * (x * (x * x)))
	elif t_1 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -5e-250)
		tmp = Float64(t_0 - Float64(Float64(x * x) * Float64(x * Float64(x * x))));
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	tmp = 0.0;
	if (t_1 <= -5e-250)
		tmp = t_0 - ((x * x) * (x * (x * x)));
	elseif (t_1 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $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-250], N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $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))) < -5.00000000000000027e-250

   1. Initial program 97.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if -5.00000000000000027e-250 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 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. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-250}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* x x) (* x (* x x))))))
   (if (<= t_1 -5e-250)
     t_2
     (if (<= t_1 0.0) (* eps (* 5.0 (pow x 4.0))) t_2))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 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-250) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - ((x * x) * (x * (x * x)))
    if (t_1 <= (-5d-250)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double t_2 = t_0 - ((x * x) * (x * (x * x)));
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - ((x * x) * (x * (x * x)))
	tmp = 0
	if t_1 <= -5e-250:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(x * x) * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (t_1 <= -5e-250)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - ((x * x) * (x * (x * x)));
	tmp = 0.0;
	if (t_1 <= -5e-250)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $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[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-250], t$95$2, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $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))) < -5.00000000000000027e-250 or 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. 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. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 97.3

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

   4. Applied rewrites 97.3%

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

   if -5.00000000000000027e-250 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.2%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-250}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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)))