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

Percentage Accurate: 88.8% → 98.9%

Time: 9.2s

Alternatives: 6

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - {\left(\sqrt[3]{\sqrt[3]{{x}^{10}}}\right)}^{3} \cdot \sqrt[3]{{x}^{5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -5e-299)
     t_1
     (if (<= t_1 0.0)
       (* eps (* 5.0 (pow x 4.0)))
       (- t_0 (* (pow (cbrt (cbrt (pow x 10.0))) 3.0) (cbrt (pow x 5.0))))))))

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 <= -5e-299) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_0 - (pow(cbrt(cbrt(pow(x, 10.0))), 3.0) * cbrt(pow(x, 5.0)));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-299) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_0 - (Math.pow(Math.cbrt(Math.cbrt(Math.pow(x, 10.0))), 3.0) * Math.cbrt(Math.pow(x, 5.0)));
	}
	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 <= -5e-299)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = Float64(t_0 - Float64((cbrt(cbrt((x ^ 10.0))) ^ 3.0) * cbrt((x ^ 5.0))));
	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[LessEqual[t$95$1, -5e-299], t$95$1, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Power[N[Power[N[Power[N[Power[x, 10.0], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[Power[x, 5.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.99999999999999956e-299

   1. Initial program 98.1%

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

   if -4.99999999999999956e-299 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

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

   1. Initial program 97.5%

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

      1. sub-neg 97.5%

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

      2. +-commutative 97.5%

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

      3. add-cube-cbrt 97.5%

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

      4. distribute-rgt-neg-in 97.5%

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

      5. fma-define 97.4%

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

      6. cbrt-unprod 97.5%

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

      7. pow-prod-up 97.5%

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

      8. metadata-eval 97.5%

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

   4. Applied egg-rr 97.5%

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

      1. fma-undefine 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. distribute-lft-neg-out 97.6%

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

      4. +-commutative 97.6%

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

      5. distribute-lft-neg-out 97.6%

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

      6. unsub-neg 97.6%

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

      7. +-commutative 97.6%

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

   6. Simplified 97.6%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.6%

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

   8. Applied egg-rr 97.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-299}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {\left(\sqrt[3]{\sqrt[3]{{x}^{10}}}\right)}^{3} \cdot \sqrt[3]{{x}^{5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \sqrt[3]{{x}^{10}} \cdot \sqrt[3]{{x}^{5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -5e-299)
     t_1
     (if (<= t_1 0.0)
       (* eps (* 5.0 (pow x 4.0)))
       (- t_0 (* (cbrt (pow x 10.0)) (cbrt (pow x 5.0))))))))

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 <= -5e-299) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_0 - (cbrt(pow(x, 10.0)) * cbrt(pow(x, 5.0)));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-299) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_0 - (Math.cbrt(Math.pow(x, 10.0)) * Math.cbrt(Math.pow(x, 5.0)));
	}
	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 <= -5e-299)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = Float64(t_0 - Float64(cbrt((x ^ 10.0)) * cbrt((x ^ 5.0))));
	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[LessEqual[t$95$1, -5e-299], t$95$1, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Power[N[Power[x, 10.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[x, 5.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.99999999999999956e-299

   1. Initial program 98.1%

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

   if -4.99999999999999956e-299 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

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

   1. Initial program 97.5%

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

      1. sub-neg 97.5%

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

      2. +-commutative 97.5%

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

      3. add-cube-cbrt 97.5%

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

      4. distribute-rgt-neg-in 97.5%

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

      5. fma-define 97.4%

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

      6. cbrt-unprod 97.5%

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

      7. pow-prod-up 97.5%

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

      8. metadata-eval 97.5%

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

   4. Applied egg-rr 97.5%

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

      1. fma-undefine 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. distribute-lft-neg-out 97.6%

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

      4. +-commutative 97.6%

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

      5. distribute-lft-neg-out 97.6%

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

      6. unsub-neg 97.6%

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

      7. +-commutative 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-299}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \sqrt[3]{{x}^{10}} \cdot \sqrt[3]{{x}^{5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-299} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-299) (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 <= -5e-299) || !(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 <= (-5d-299)) .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 <= -5e-299) || !(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 <= -5e-299) 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 <= -5e-299) || !(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 <= -5e-299) || ~((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, -5e-299], 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)) < -4.99999999999999956e-299 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.8%

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

   if -4.99999999999999956e-299 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-299} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-50} \lor \neg \left(x \leq 3.5 \cdot 10^{-53}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.65e-50) (not (<= x 3.5e-53)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.65e-50) || !(x <= 3.5e-53)) {
		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 <= (-1.65d-50)) .or. (.not. (x <= 3.5d-53))) 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 <= -1.65e-50) || !(x <= 3.5e-53)) {
		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 <= -1.65e-50) or not (x <= 3.5e-53):
		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 <= -1.65e-50) || !(x <= 3.5e-53))
		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 <= -1.65e-50) || ~((x <= 3.5e-53)))
		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, -1.65e-50], N[Not[LessEqual[x, 3.5e-53]], $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 < -1.6499999999999999e-50 or 3.49999999999999993e-53 < x

   1. Initial program 43.2%

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

   3. Taylor expanded in x around inf 91.6%

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

      1. *-commutative 91.6%

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

      2. distribute-rgt1-in 91.6%

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

      3. metadata-eval 91.6%

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

      4. *-commutative 91.6%

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

      5. associate-*r* 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in eps around 0 91.7%

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

   if -1.6499999999999999e-50 < x < 3.49999999999999993e-53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-50} \lor \neg \left(x \leq 3.5 \cdot 10^{-53}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-52}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.5e-52)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 1.45e-52) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.5e-52) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.45e-52) {
		tmp = pow(eps, 5.0);
	} 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) :: tmp
    if (x <= (-9.5d-52)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.45d-52) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9.5e-52) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.45e-52) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9.5e-52:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.45e-52:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.5e-52)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.45e-52)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9.5e-52)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.45e-52)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.5e-52], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-52], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.50000000000000007e-52

   1. Initial program 51.1%

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

   3. Taylor expanded in x around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. distribute-rgt1-in 86.9%

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

      3. metadata-eval 86.9%

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

      4. *-commutative 86.9%

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

      5. associate-*r* 87.0%

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

   5. Simplified 87.0%

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

   6. Taylor expanded in eps around 0 87.1%

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

   if -9.50000000000000007e-52 < x < 1.4500000000000001e-52

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1.4500000000000001e-52 < x

   1. Initial program 35.3%

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

   3. Taylor expanded in x around inf 96.3%

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

      1. *-commutative 96.3%

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

      2. distribute-rgt1-in 96.3%

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

      3. metadata-eval 96.3%

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

      4. *-commutative 96.3%

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

      5. associate-*r* 96.4%

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

   5. Simplified 96.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-52}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.9% 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 91.6%

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

3. Taylor expanded in x around 0 90.3%

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

4. Final simplification 90.3%

   \[\leadsto {\varepsilon}^{5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))