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

Percentage Accurate: 88.5% → 99.1%

Time: 9.4s

Alternatives: 15

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 + {x}^{5}\\ t_3 := \frac{t\_1}{t\_2} \cdot t\_2\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (pow x 5.0)))
        (t_3 (* (/ t_1 t_2) t_2)))
   (if (<= t_1 -2e-301)
     t_3
     (if (<= t_1 0.0) (* (* (pow x 4.0) 5.0) eps) t_3))))

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 + pow(x, 5.0);
	double t_3 = (t_1 / t_2) * t_2;
	double tmp;
	if (t_1 <= -2e-301) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = (eps + x) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 + (x ** 5.0d0)
    t_3 = (t_1 / t_2) * t_2
    if (t_1 <= (-2d-301)) then
        tmp = t_3
    else if (t_1 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((eps + x), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double t_2 = t_0 + Math.pow(x, 5.0);
	double t_3 = (t_1 / t_2) * t_2;
	double tmp;
	if (t_1 <= -2e-301) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((eps + x), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 + math.pow(x, 5.0)
	t_3 = (t_1 / t_2) * t_2
	tmp = 0
	if t_1 <= -2e-301:
		tmp = t_3
	elif t_1 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = t_3
	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 + (x ^ 5.0))
	t_3 = Float64(Float64(t_1 / t_2) * t_2)
	tmp = 0.0
	if (t_1 <= -2e-301)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps + x) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 + (x ^ 5.0);
	t_3 = (t_1 / t_2) * t_2;
	tmp = 0.0;
	if (t_1 <= -2e-301)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = t_3;
	end
	tmp_2 = 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[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-301], t$95$3, If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$3]]]]]]

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))) < -2.00000000000000013e-301 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. difference-of-squares N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

   4. Applied rewrites 98.0%

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

   if -2.00000000000000013e-301 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 82.5%

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

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-301}:\\ \;\;\;\;\frac{{\left(\varepsilon + x\right)}^{5} - {x}^{5}}{{\left(\varepsilon + x\right)}^{5} + {x}^{5}} \cdot \left({\left(\varepsilon + x\right)}^{5} + {x}^{5}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\varepsilon + x\right)}^{5} - {x}^{5}}{{\left(\varepsilon + x\right)}^{5} + {x}^{5}} \cdot \left({\left(\varepsilon + x\right)}^{5} + {x}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-301)
     t_0
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-301) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((eps + x) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-2d-301)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((eps + x), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-301) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((eps + x), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -2e-301:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-301)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((eps + x) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -2e-301)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-301], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

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))) < -2.00000000000000013e-301 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.0%

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

   if -2.00000000000000013e-301 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 82.5%

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

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-301}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + \frac{\mathsf{fma}\left(5, x, \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, -10, -10\right) \cdot x\right) \cdot x\right) \cdot \frac{-1}{\varepsilon}\right)}{\varepsilon}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-301)
     (*
      (pow eps 5.0)
      (+
       1.0
       (/
        (fma 5.0 x (* (* (* (fma (/ x eps) -10.0 -10.0) x) x) (/ -1.0 eps)))
        eps)))
     (if (<= t_0 0.0)
       (* (* (pow x 4.0) 5.0) eps)
       (*
        (+ (/ (fma 5.0 x (/ (* (* x x) -10.0) (- eps))) eps) 1.0)
        (pow eps 5.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-301) {
		tmp = pow(eps, 5.0) * (1.0 + (fma(5.0, x, (((fma((x / eps), -10.0, -10.0) * x) * x) * (-1.0 / eps))) / eps));
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = ((fma(5.0, x, (((x * x) * -10.0) / -eps)) / eps) + 1.0) * pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-301)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(fma(5.0, x, Float64(Float64(Float64(fma(Float64(x / eps), -10.0, -10.0) * x) * x) * Float64(-1.0 / eps))) / eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = Float64(Float64(Float64(fma(5.0, x, Float64(Float64(Float64(x * x) * -10.0) / Float64(-eps))) / eps) + 1.0) * (eps ^ 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-301], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(N[(5.0 * x + N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * -10.0 + -10.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(5.0 * x + N[(N[(N[(x * x), $MachinePrecision] * -10.0), $MachinePrecision] / (-eps)), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2.00000000000000013e-301

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\mathsf{fma}\left(x \cdot x, -10, \frac{{x}^{3} \cdot 10}{-\varepsilon}\right)}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.5%

      \[\leadsto \left(\frac{\mathsf{fma}\left(5, x, \frac{\mathsf{fma}\left(\frac{x}{\varepsilon}, -10, -10\right) \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.5%

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

   if -2.00000000000000013e-301 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 82.5%

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

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 97.6%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-301}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + \frac{\mathsf{fma}\left(5, x, \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, -10, -10\right) \cdot x\right) \cdot x\right) \cdot \frac{-1}{\varepsilon}\right)}{\varepsilon}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{x}{\varepsilon}, -10, -10\right)}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-301)
     (*
      (+
       (/ (fma 5.0 x (/ (* (* x x) (fma (/ x eps) -10.0 -10.0)) (- eps))) eps)
       1.0)
      (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* (pow x 4.0) 5.0) eps)
       (*
        (+ (/ (fma 5.0 x (/ (* (* x x) -10.0) (- eps))) eps) 1.0)
        (pow eps 5.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-301) {
		tmp = ((fma(5.0, x, (((x * x) * fma((x / eps), -10.0, -10.0)) / -eps)) / eps) + 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = ((fma(5.0, x, (((x * x) * -10.0) / -eps)) / eps) + 1.0) * pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-301)
		tmp = Float64(Float64(Float64(fma(5.0, x, Float64(Float64(Float64(x * x) * fma(Float64(x / eps), -10.0, -10.0)) / Float64(-eps))) / eps) + 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = Float64(Float64(Float64(fma(5.0, x, Float64(Float64(Float64(x * x) * -10.0) / Float64(-eps))) / eps) + 1.0) * (eps ^ 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-301], N[(N[(N[(N[(5.0 * x + N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x / eps), $MachinePrecision] * -10.0 + -10.0), $MachinePrecision]), $MachinePrecision] / (-eps)), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(5.0 * x + N[(N[(N[(x * x), $MachinePrecision] * -10.0), $MachinePrecision] / (-eps)), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -2.00000000000000013e-301

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\mathsf{fma}\left(x \cdot x, -10, \frac{{x}^{3} \cdot 10}{-\varepsilon}\right)}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.5%

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

   if -2.00000000000000013e-301 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 82.5%

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

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 97.6%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-301}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{x}{\varepsilon}, -10, -10\right)}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(5, x, \frac{\left(x \cdot x\right) \cdot -10}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1
         (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (* eps eps)) eps)))
   (if (<= t_0 -2e-301)
     t_1
     (if (<= t_0 0.0) (* (* (* (* (* x x) x) x) 5.0) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -2e-301) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((((x * x) * x) * x) * 5.0) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * Float64(eps * eps)) * eps)
	tmp = 0.0
	if (t_0 <= -2e-301)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * 5.0) * eps);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-301], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

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))) < -2.00000000000000013e-301 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.0%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.5%

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

   10. Applied rewrites 88.2%

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

   if -2.00000000000000013e-301 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-sub N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   6. 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 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-301}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1
         (* (* (* eps eps) eps) (fma (fma 5.0 x eps) eps (* (* x x) 10.0)))))
   (if (<= t_0 -2e-301)
     t_1
     (if (<= t_0 0.0) (* (* (* (* (* x x) x) x) 5.0) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * eps) * fma(fma(5.0, x, eps), eps, ((x * x) * 10.0));
	double tmp;
	if (t_0 <= -2e-301) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((((x * x) * x) * x) * 5.0) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * eps) * fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)))
	tmp = 0.0
	if (t_0 <= -2e-301)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * 5.0) * eps);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-301], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

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))) < -2.00000000000000013e-301 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.0%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.5%

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

   10. Applied rewrites 88.2%

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

   if -2.00000000000000013e-301 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-sub N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   6. 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 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-301}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon \cdot \varepsilon}{x}, 10, 4 \cdot \varepsilon\right) + \varepsilon\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-58)
   (* (+ (fma (/ (* eps eps) x) 10.0 (* 4.0 eps)) eps) (pow x 4.0))
   (if (<= x 4.5e-33)
     (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (pow eps 3.0))
     (* (* (fma (/ eps x) 10.0 5.0) eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-58) {
		tmp = (fma(((eps * eps) / x), 10.0, (4.0 * eps)) + eps) * pow(x, 4.0);
	} else if (x <= 4.5e-33) {
		tmp = fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * pow(eps, 3.0);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-58)
		tmp = Float64(Float64(fma(Float64(Float64(eps * eps) / x), 10.0, Float64(4.0 * eps)) + eps) * (x ^ 4.0));
	elseif (x <= 4.5e-33)
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * (eps ^ 3.0));
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-58], N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision] * 10.0 + N[(4.0 * eps), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-33], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999995e-58

   1. Initial program 33.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if -1.44999999999999995e-58 < x < 4.49999999999999991e-33

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

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

   if 4.49999999999999991e-33 < x

   1. Initial program 13.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.3%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon \cdot \varepsilon}{x}, 10, 4 \cdot \varepsilon\right) + \varepsilon\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (fma (/ eps x) 10.0 5.0) eps) (pow x 4.0))))
   (if (<= x -1.45e-58)
     t_0
     (if (<= x 4.5e-33)
       (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (pow eps 3.0))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = (fma((eps / x), 10.0, 5.0) * eps) * pow(x, 4.0);
	double tmp;
	if (x <= -1.45e-58) {
		tmp = t_0;
	} else if (x <= 4.5e-33) {
		tmp = fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * pow(eps, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * (x ^ 4.0))
	tmp = 0.0
	if (x <= -1.45e-58)
		tmp = t_0;
	elseif (x <= 4.5e-33)
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * (eps ^ 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-58], t$95$0, If[LessEqual[x, 4.5e-33], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999995e-58 or 4.49999999999999991e-33 < x

   1. Initial program 27.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 93.4%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 93.4%

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

   if -1.44999999999999995e-58 < x < 4.49999999999999991e-33

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{10}{x}, \varepsilon \cdot \varepsilon, 5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-58)
   (* (pow x 3.0) (fma (* eps x) 5.0 (* (* eps eps) 10.0)))
   (if (<= x 4.5e-33)
     (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (pow eps 3.0))
     (* (* (fma (/ 10.0 x) (* eps eps) (* 5.0 eps)) (* x x)) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-58) {
		tmp = pow(x, 3.0) * fma((eps * x), 5.0, ((eps * eps) * 10.0));
	} else if (x <= 4.5e-33) {
		tmp = fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * pow(eps, 3.0);
	} else {
		tmp = (fma((10.0 / x), (eps * eps), (5.0 * eps)) * (x * x)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-58)
		tmp = Float64((x ^ 3.0) * fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)));
	elseif (x <= 4.5e-33)
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * (eps ^ 3.0));
	else
		tmp = Float64(Float64(fma(Float64(10.0 / x), Float64(eps * eps), Float64(5.0 * eps)) * Float64(x * x)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-58], N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-33], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(10.0 / x), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999995e-58

   1. Initial program 33.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.8%

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

   if -1.44999999999999995e-58 < x < 4.49999999999999991e-33

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

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

   if 4.49999999999999991e-33 < x

   1. Initial program 13.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.3%

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

   7. Applied rewrites 99.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{10}{x}, \varepsilon \cdot \varepsilon, 5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{10}{x}, \varepsilon \cdot \varepsilon, 5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-58)
   (* (pow x 3.0) (fma (* eps x) 5.0 (* (* eps eps) 10.0)))
   (if (<= x 4.5e-33)
     (* (fma (* x x) 10.0 (* (fma 5.0 x eps) eps)) (pow eps 3.0))
     (* (* (fma (/ 10.0 x) (* eps eps) (* 5.0 eps)) (* x x)) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-58) {
		tmp = pow(x, 3.0) * fma((eps * x), 5.0, ((eps * eps) * 10.0));
	} else if (x <= 4.5e-33) {
		tmp = fma((x * x), 10.0, (fma(5.0, x, eps) * eps)) * pow(eps, 3.0);
	} else {
		tmp = (fma((10.0 / x), (eps * eps), (5.0 * eps)) * (x * x)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-58)
		tmp = Float64((x ^ 3.0) * fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)));
	elseif (x <= 4.5e-33)
		tmp = Float64(fma(Float64(x * x), 10.0, Float64(fma(5.0, x, eps) * eps)) * (eps ^ 3.0));
	else
		tmp = Float64(Float64(fma(Float64(10.0 / x), Float64(eps * eps), Float64(5.0 * eps)) * Float64(x * x)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-58], N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-33], N[(N[(N[(x * x), $MachinePrecision] * 10.0 + N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(10.0 / x), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999995e-58

   1. Initial program 33.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.8%

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

   if -1.44999999999999995e-58 < x < 4.49999999999999991e-33

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.7%

       \[\leadsto {\varepsilon}^{3} \cdot \mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \]

   if 4.49999999999999991e-33 < x

   1. Initial program 13.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.3%

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

   7. Applied rewrites 99.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 10, \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot {\varepsilon}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{10}{x}, \varepsilon \cdot \varepsilon, 5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{10}{x}, \varepsilon \cdot \varepsilon, 5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-58)
   (* (pow x 3.0) (fma (* eps x) 5.0 (* (* eps eps) 10.0)))
   (if (<= x 4.5e-33)
     (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (* eps eps)) eps)
     (* (* (fma (/ 10.0 x) (* eps eps) (* 5.0 eps)) (* x x)) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-58) {
		tmp = pow(x, 3.0) * fma((eps * x), 5.0, ((eps * eps) * 10.0));
	} else if (x <= 4.5e-33) {
		tmp = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * (eps * eps)) * eps;
	} else {
		tmp = (fma((10.0 / x), (eps * eps), (5.0 * eps)) * (x * x)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-58)
		tmp = Float64((x ^ 3.0) * fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)));
	elseif (x <= 4.5e-33)
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * Float64(eps * eps)) * eps);
	else
		tmp = Float64(Float64(fma(Float64(10.0 / x), Float64(eps * eps), Float64(5.0 * eps)) * Float64(x * x)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-58], N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-33], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(10.0 / x), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999995e-58

   1. Initial program 33.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.8%

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

   if -1.44999999999999995e-58 < x < 4.49999999999999991e-33

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.6%

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

   if 4.49999999999999991e-33 < x

   1. Initial program 13.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.3%

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

   7. Applied rewrites 99.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;{x}^{3} \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{10}{x}, \varepsilon \cdot \varepsilon, 5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\frac{10}{x}, \varepsilon \cdot \varepsilon, 5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (fma (/ 10.0 x) (* eps eps) (* 5.0 eps)) (* x x)) (* x x))))
   (if (<= x -1.45e-58)
     t_0
     (if (<= x 4.5e-33)
       (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (* eps eps)) eps)
       t_0))))

C

double code(double x, double eps) {
	double t_0 = (fma((10.0 / x), (eps * eps), (5.0 * eps)) * (x * x)) * (x * x);
	double tmp;
	if (x <= -1.45e-58) {
		tmp = t_0;
	} else if (x <= 4.5e-33) {
		tmp = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(fma(Float64(10.0 / x), Float64(eps * eps), Float64(5.0 * eps)) * Float64(x * x)) * Float64(x * x))
	tmp = 0.0
	if (x <= -1.45e-58)
		tmp = t_0;
	elseif (x <= 4.5e-33)
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * Float64(eps * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(10.0 / x), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-58], t$95$0, If[LessEqual[x, 4.5e-33], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999995e-58 or 4.49999999999999991e-33 < x

   1. Initial program 27.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 93.4%

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

   7. Applied rewrites 93.2%

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

   if -1.44999999999999995e-58 < x < 4.49999999999999991e-33

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.6%

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

Alternative 13: 82.0% accurate, 8.0× speedup

Math

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

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(Float64(x * x) * 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[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. div-sub N/A

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

   4. frac-2neg N/A

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

   5. frac-2neg N/A

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

   6. sub-div N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 85.5

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

4. Applied rewrites 85.5%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
6. 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 81.8

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

7. Applied rewrites 81.8%

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

9. Applied rewrites 81.8%

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

11. Applied rewrites 81.8%

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

12. Final simplification 81.8%

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

Alternative 14: 82.0% 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 85.6%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. div-sub N/A

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

   4. frac-2neg N/A

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

   5. frac-2neg N/A

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

   6. sub-div N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 85.5

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

4. Applied rewrites 85.5%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
6. 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 81.8

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

7. Applied rewrites 81.8%

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

9. Applied rewrites 81.7%

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

10. Final simplification 81.7%

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

Alternative 15: 70.6% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in eps around -inf

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

5. Applied rewrites 72.5%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(5, x, \frac{\mathsf{fma}\left(x \cdot x, -10, \frac{{x}^{3} \cdot 10}{-\varepsilon}\right)}{-\varepsilon}\right)}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 67.4%

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

9. Taylor expanded in eps around inf

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

11. Applied rewrites 67.5%

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

12. Final simplification 67.5%

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

Reproduce

herbie shell --seed 2024270 
(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)))