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

Percentage Accurate: 88.4% → 99.4%

Time: 7.9s

Alternatives: 7

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t_0 - {x}^{5}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-314} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{{x}^{5}}\right)}^{2}, \sqrt[3]{-{x}^{5}}, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (or (<= t_1 -1e-314) (not (<= t_1 0.0)))
     (fma (pow (cbrt (pow x 5.0)) 2.0) (cbrt (- (pow x 5.0))) t_0)
     (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if ((t_1 <= -1e-314) || !(t_1 <= 0.0)) {
		tmp = fma(pow(cbrt(pow(x, 5.0)), 2.0), cbrt(-pow(x, 5.0)), t_0);
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if ((t_1 <= -1e-314) || !(t_1 <= 0.0))
		tmp = fma((cbrt((x ^ 5.0)) ^ 2.0), cbrt(Float64(-(x ^ 5.0))), t_0);
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-314], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[(-N[Power[x, 5.0], $MachinePrecision]), 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.9999999996e-315 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.1%

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

      1. sub-neg 98.1%

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

      2. +-commutative 98.1%

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

      3. add-cube-cbrt 98.2%

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

      4. fma-def 98.2%

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

   3. Applied egg-rr 98.2%

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

      1. pow1/3 91.3%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(-{x}^{5}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{-{x}^{5}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      2. pow1/3 91.2%

         \[\leadsto \mathsf{fma}\left({\left(-{x}^{5}\right)}^{0.3333333333333333} \cdot \color{blue}{{\left(-{x}^{5}\right)}^{0.3333333333333333}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      3. pow-prod-up 91.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(-{x}^{5}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      4. add-sqr-sqrt 91.2%

         \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\sqrt{-{x}^{5}} \cdot \sqrt{-{x}^{5}}\right)}}^{\left(0.3333333333333333 + 0.3333333333333333\right)}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      5. sqrt-unprod 95.9%

         \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\sqrt{\left(-{x}^{5}\right) \cdot \left(-{x}^{5}\right)}\right)}}^{\left(0.3333333333333333 + 0.3333333333333333\right)}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      6. sqr-neg 95.9%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt{\color{blue}{{x}^{5} \cdot {x}^{5}}}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      7. sqrt-unprod 92.8%

         \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\sqrt{{x}^{5}} \cdot \sqrt{{x}^{5}}\right)}}^{\left(0.3333333333333333 + 0.3333333333333333\right)}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      8. add-sqr-sqrt 92.8%

         \[\leadsto \mathsf{fma}\left({\color{blue}{\left({x}^{5}\right)}}^{\left(0.3333333333333333 + 0.3333333333333333\right)}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      9. metadata-eval 92.8%

         \[\leadsto \mathsf{fma}\left({\left({x}^{5}\right)}^{\color{blue}{0.6666666666666666}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

   5. Applied egg-rr 92.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({x}^{5}\right)}^{0.6666666666666666}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]
   6. Step-by-step derivation

      1. metadata-eval 92.8%

         \[\leadsto \mathsf{fma}\left({\left({x}^{5}\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      2. pow-sqr 92.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({x}^{5}\right)}^{0.3333333333333333} \cdot {\left({x}^{5}\right)}^{0.3333333333333333}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      3. unpow1/3 92.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{{x}^{5}}} \cdot {\left({x}^{5}\right)}^{0.3333333333333333}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

      4. unpow1/3 98.2%

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

      5. unpow2 98.2%

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

   7. Simplified 98.2%

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

   if -9.9999999996e-315 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 87.3%

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

      1. flip-- 13.7%

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

      2. pow-prod-up 12.4%

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

      3. metadata-eval 12.4%

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

      4. pow-prod-up 13.7%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - \color{blue}{{x}^{\left(5 + 5\right)}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

      5. metadata-eval 13.7%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - {x}^{\color{blue}{10}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

   3. Applied egg-rr 13.7%

      \[\leadsto \color{blue}{\frac{{\left(x + \varepsilon\right)}^{10} - {x}^{10}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}}} \]

   4. Taylor expanded in eps around 0 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-314} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{{x}^{5}}\right)}^{2}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-314} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-314) (not (<= t_0 0.0)))
     t_0
     (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-314) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	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 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-1d-314)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-314) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -1e-314) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -1e-314) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -1e-314) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-314], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.9999999996e-315 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.1%

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

   if -9.9999999996e-315 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 87.3%

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

      1. flip-- 13.7%

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

      2. pow-prod-up 12.4%

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

      3. metadata-eval 12.4%

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

      4. pow-prod-up 13.7%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - \color{blue}{{x}^{\left(5 + 5\right)}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

      5. metadata-eval 13.7%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - {x}^{\color{blue}{10}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

   3. Applied egg-rr 13.7%

      \[\leadsto \color{blue}{\frac{{\left(x + \varepsilon\right)}^{10} - {x}^{10}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}}} \]

   4. Taylor expanded in eps around 0 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-314} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \sqrt{{x}^{8} \cdot 25}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-41)
   (* eps (sqrt (* (pow x 8.0) 25.0)))
   (if (<= x 1.06e-60) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-41) {
		tmp = eps * sqrt((pow(x, 8.0) * 25.0));
	} else if (x <= 1.06e-60) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5d-41)) then
        tmp = eps * sqrt(((x ** 8.0d0) * 25.0d0))
    else if (x <= 1.06d-60) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5e-41) {
		tmp = eps * Math.sqrt((Math.pow(x, 8.0) * 25.0));
	} else if (x <= 1.06e-60) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5e-41:
		tmp = eps * math.sqrt((math.pow(x, 8.0) * 25.0))
	elif x <= 1.06e-60:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e-41)
		tmp = Float64(eps * sqrt(Float64((x ^ 8.0) * 25.0)));
	elseif (x <= 1.06e-60)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5e-41)
		tmp = eps * sqrt(((x ^ 8.0) * 25.0));
	elseif (x <= 1.06e-60)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e-41], N[(eps * N[Sqrt[N[(N[Power[x, 8.0], $MachinePrecision] * 25.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e-60], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999996e-41

   1. Initial program 32.1%

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

      1. flip-- 32.1%

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

      2. pow-prod-up 22.5%

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

      3. metadata-eval 22.5%

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

      4. pow-prod-up 32.4%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - \color{blue}{{x}^{\left(5 + 5\right)}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

      5. metadata-eval 32.4%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - {x}^{\color{blue}{10}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

   3. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\frac{{\left(x + \varepsilon\right)}^{10} - {x}^{10}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}}} \]

   4. Taylor expanded in eps around 0 94.3%

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

      1. distribute-lft1-in 94.3%

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

      2. metadata-eval 94.3%

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

   6. Simplified 94.3%

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

      1. add-sqr-sqrt 93.9%

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

      2. sqrt-unprod 94.3%

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

      3. swap-sqr 94.0%

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

      4. metadata-eval 94.0%

         \[\leadsto \varepsilon \cdot \sqrt{\color{blue}{25} \cdot \left({x}^{4} \cdot {x}^{4}\right)} \]

      5. pow-prod-up 94.3%

         \[\leadsto \varepsilon \cdot \sqrt{25 \cdot \color{blue}{{x}^{\left(4 + 4\right)}}} \]

      6. metadata-eval 94.3%

         \[\leadsto \varepsilon \cdot \sqrt{25 \cdot {x}^{\color{blue}{8}}} \]

   8. Applied egg-rr 94.3%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{25 \cdot {x}^{8}}} \]
   9. Step-by-step derivation

      1. *-commutative 94.3%

         \[\leadsto \varepsilon \cdot \sqrt{\color{blue}{{x}^{8} \cdot 25}} \]

   10. Simplified 94.3%

       \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]

   if -4.9999999999999996e-41 < x < 1.06e-60

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

   if 1.06e-60 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf 94.7%

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

      1. distribute-lft1-in 94.7%

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

      2. metadata-eval 94.7%

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

      3. associate-*l* 94.7%

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

   4. Simplified 94.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\varepsilon \cdot \sqrt{{x}^{8} \cdot 25}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-46} \lor \neg \left(x \leq 1.06 \cdot 10^{-60}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.5e-46) (not (<= x 1.06e-60)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.5e-46) || !(x <= 1.06e-60)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.5d-46)) .or. (.not. (x <= 1.06d-60))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.5e-46) || !(x <= 1.06e-60)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.5e-46) or not (x <= 1.06e-60):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.5e-46) || !(x <= 1.06e-60))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.5e-46) || ~((x <= 1.06e-60)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.5e-46], N[Not[LessEqual[x, 1.06e-60]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.49999999999999996e-46 or 1.06e-60 < x

   1. Initial program 46.1%

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

   2. Taylor expanded in x around inf 94.5%

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

      1. distribute-lft1-in 94.5%

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

      2. metadata-eval 94.5%

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

      3. associate-*l* 94.5%

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

   4. Simplified 94.5%

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

   if -2.49999999999999996e-46 < x < 1.06e-60

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-46} \lor \neg \left(x \leq 1.06 \cdot 10^{-60}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-42)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 6e-61) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-42) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 6e-61) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1d-42)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 6d-61) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1e-42) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 6e-61) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1e-42:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 6e-61:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-42)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 6e-61)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1e-42)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 6e-61)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-42], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-61], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000004e-42

   1. Initial program 32.1%

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

      1. flip-- 32.1%

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

      2. pow-prod-up 22.5%

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

      3. metadata-eval 22.5%

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

      4. pow-prod-up 32.4%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - \color{blue}{{x}^{\left(5 + 5\right)}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

      5. metadata-eval 32.4%

         \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - {x}^{\color{blue}{10}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

   3. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\frac{{\left(x + \varepsilon\right)}^{10} - {x}^{10}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}}} \]

   4. Taylor expanded in eps around 0 94.3%

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

      1. distribute-lft1-in 94.3%

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

      2. metadata-eval 94.3%

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

   6. Simplified 94.3%

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

   if -1.00000000000000004e-42 < x < 6.00000000000000024e-61

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

   if 6.00000000000000024e-61 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf 94.7%

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

      1. distribute-lft1-in 94.7%

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

      2. metadata-eval 94.7%

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

      3. associate-*l* 94.7%

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

   4. Simplified 94.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 6: 87.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return eps ^ 5.0
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Power[eps, 5.0], $MachinePrecision]

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in x around 0 88.3%

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

3. Final simplification 88.3%

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

Alternative 7: 70.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 89.1%

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

   1. flip-- 22.2%

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

   2. pow-prod-up 21.2%

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

   3. metadata-eval 21.2%

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

   4. pow-prod-up 22.2%

      \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - \color{blue}{{x}^{\left(5 + 5\right)}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

   5. metadata-eval 22.2%

      \[\leadsto \frac{{\left(x + \varepsilon\right)}^{10} - {x}^{\color{blue}{10}}}{{\left(x + \varepsilon\right)}^{5} + {x}^{5}} \]

3. Applied egg-rr 22.2%

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

   1. metadata-eval 22.2%

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

   2. pow-sqr 21.2%

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

   3. metadata-eval 21.2%

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

   4. pow-sqr 22.2%

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

   5. flip-- 89.1%

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

   6. sqr-pow 41.1%

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

   7. cancel-sign-sub-inv 41.1%

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

   8. +-commutative 41.1%

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

   9. metadata-eval 41.1%

      \[\leadsto {\left(\varepsilon + x\right)}^{5} + \left(-{x}^{\color{blue}{2.5}}\right) \cdot {x}^{\left(\frac{5}{2}\right)} \]

   10. metadata-eval 41.1%

       \[\leadsto {\left(\varepsilon + x\right)}^{5} + \left(-{x}^{2.5}\right) \cdot {x}^{\color{blue}{2.5}} \]

5. Applied egg-rr 41.1%

   \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{5} + \left(-{x}^{2.5}\right) \cdot {x}^{2.5}} \]

6. Taylor expanded in eps around 0 73.3%

   \[\leadsto \color{blue}{{x}^{5} + -1 \cdot {x}^{5}} \]
7. Step-by-step derivation

   1. distribute-rgt1-in 73.3%

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

   2. metadata-eval 73.3%

      \[\leadsto \color{blue}{0} \cdot {x}^{5} \]

   3. mul0-lft 73.3%

      \[\leadsto \color{blue}{0} \]

8. Simplified 73.3%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 73.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023171 
(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)))