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

Percentage Accurate: 88.7% → 98.8%

Time: 10.9s

Alternatives: 8

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := \sqrt{-{x}^{5}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t\_2, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (sqrt (- (pow x 5.0)))))
   (if (<= t_1 -1e-314)
     (fma t_2 t_2 t_0)
     (if (<= t_1 0.0) (* (pow x 4.0) (* eps 5.0)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = sqrt(-pow(x, 5.0));
	double tmp;
	if (t_1 <= -1e-314) {
		tmp = fma(t_2, t_2, t_0);
	} else if (t_1 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = sqrt(Float64(-(x ^ 5.0)))
	tmp = 0.0
	if (t_1 <= -1e-314)
		tmp = fma(t_2, t_2, t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = t_1;
	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]}, Block[{t$95$2 = N[Sqrt[(-N[Power[x, 5.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[t$95$1, -1e-314], N[(t$95$2 * t$95$2 + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9999999996e-315

   1. Initial program 96.7%

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

      1. sub-neg 96.7%

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

      2. +-commutative 96.7%

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

      3. add-sqr-sqrt 97.0%

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

      4. fma-define 96.7%

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

   4. Applied egg-rr 96.7%

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

   if -9.9999999996e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

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

   1. Initial program 99.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-314} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

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
     (* (pow x 4.0) (* eps 5.0)))))

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 = pow(x, 4.0) * (eps * 5.0);
	}
	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 = (x ** 4.0d0) * (eps * 5.0d0)
    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 = Math.pow(x, 4.0) * (eps * 5.0);
	}
	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 = math.pow(x, 4.0) * (eps * 5.0)
	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((x ^ 4.0) * Float64(eps * 5.0));
	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 = (x ^ 4.0) * (eps * 5.0);
	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[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

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))) < -9.9999999996e-315 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.8%

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

   if -9.9999999996e-315 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.5%

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

Alternative 3: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-46} \lor \neg \left(x \leq 6.8 \cdot 10^{-87}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -9.8e-46) (not (<= x 6.8e-87)))
   (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -9.8e-46) || !(x <= 6.8e-87)) {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} 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 <= (-9.8d-46)) .or. (.not. (x <= 6.8d-87))) then
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / x))))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -9.8e-46) || !(x <= 6.8e-87)) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -9.8e-46) or not (x <= 6.8e-87):
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -9.8e-46) || !(x <= 6.8e-87))
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -9.8e-46) || ~((x <= 6.8e-87)))
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -9.8e-46], N[Not[LessEqual[x, 6.8e-87]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.8000000000000002e-46 or 6.7999999999999997e-87 < x

   1. Initial program 41.2%

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

   3. Taylor expanded in x around -inf 97.6%

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

      1. +-commutative 97.6%

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

      2. associate-+r+ 97.6%

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

      3. mul-1-neg 97.6%

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

      4. unsub-neg 97.6%

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

      5. distribute-rgt1-in 97.6%

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

      6. metadata-eval 97.6%

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

      7. *-commutative 97.6%

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

   5. Simplified 97.6%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]

   6. Taylor expanded in eps around 0 97.6%

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

   if -9.8000000000000002e-46 < x < 6.7999999999999997e-87

   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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-46} \lor \neg \left(x \leq 6.8 \cdot 10^{-87}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-45} \lor \neg \left(x \leq 6.8 \cdot 10^{-87}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.2e-45) (not (<= x 6.8e-87)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.2e-45) || !(x <= 6.8e-87)) {
		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 <= (-4.2d-45)) .or. (.not. (x <= 6.8d-87))) 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 <= -4.2e-45) || !(x <= 6.8e-87)) {
		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 <= -4.2e-45) or not (x <= 6.8e-87):
		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 <= -4.2e-45) || !(x <= 6.8e-87))
		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 <= -4.2e-45) || ~((x <= 6.8e-87)))
		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, -4.2e-45], N[Not[LessEqual[x, 6.8e-87]], $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 < -4.1999999999999999e-45 or 6.7999999999999997e-87 < x

   1. Initial program 41.2%

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

   3. Taylor expanded in x around inf 95.8%

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

      1. *-commutative 95.8%

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

      2. distribute-rgt1-in 95.8%

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

      3. metadata-eval 95.8%

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

      4. *-commutative 95.8%

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

      5. associate-*r* 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in eps around 0 95.8%

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

   if -4.1999999999999999e-45 < x < 6.7999999999999997e-87

   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 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-45} \lor \neg \left(x \leq 6.8 \cdot 10^{-87}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-46}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-87}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.8e-46)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 6.8e-87) (pow eps 5.0) (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.8e-46) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 6.8e-87) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 4.0) * (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 <= (-9.8d-46)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 6.8d-87) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9.8e-46) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 6.8e-87) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9.8e-46:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 6.8e-87:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.8e-46)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 6.8e-87)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9.8e-46)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 6.8e-87)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.8e-46], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-87], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000002e-46

   1. Initial program 27.2%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. *-commutative 93.6%

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

      2. distribute-rgt1-in 93.6%

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

      3. metadata-eval 93.6%

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

      4. *-commutative 93.6%

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

      5. associate-*r* 93.5%

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

   5. Simplified 93.5%

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

   6. Taylor expanded in eps around 0 93.7%

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

   if -9.8000000000000002e-46 < x < 6.7999999999999997e-87

   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 6.7999999999999997e-87 < x

   1. Initial program 52.6%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. distribute-rgt1-in 97.7%

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

      2. metadata-eval 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-46}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-87}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{-44}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-87}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.82e-44)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 5e-87) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.82e-44) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 5e-87) {
		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 <= (-1.82d-44)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 5d-87) 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 <= -1.82e-44) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 5e-87) {
		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 <= -1.82e-44:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 5e-87:
		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 <= -1.82e-44)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 5e-87)
		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 <= -1.82e-44)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 5e-87)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.82e-44], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-87], 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 < -1.8200000000000001e-44

   1. Initial program 27.2%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. *-commutative 93.6%

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

      2. distribute-rgt1-in 93.6%

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

      3. metadata-eval 93.6%

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

      4. *-commutative 93.6%

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

      5. associate-*r* 93.5%

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

   5. Simplified 93.5%

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

   6. Taylor expanded in eps around 0 93.7%

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

   if -1.8200000000000001e-44 < x < 5.00000000000000042e-87

   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 5.00000000000000042e-87 < x

   1. Initial program 52.6%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. *-commutative 97.7%

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

      2. distribute-rgt1-in 97.7%

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

      3. metadata-eval 97.7%

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

      4. *-commutative 97.7%

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

      5. associate-*r* 97.5%

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

   5. Simplified 97.5%

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

Alternative 7: 87.7% 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 86.7%

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

3. Taylor expanded in x around 0 85.7%

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

Alternative 8: 70.8% 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 86.7%

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

   1. sub-neg 86.7%

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

   2. +-commutative 86.7%

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

   3. sqr-pow 35.8%

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

   4. distribute-rgt-neg-in 35.8%

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

   5. fma-define 35.2%

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

   6. metadata-eval 35.2%

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

   7. metadata-eval 35.2%

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

4. Applied egg-rr 35.2%

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

5. Taylor expanded in x around -inf 0.0%

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

   1. mul-1-neg 0.0%

      \[\leadsto \color{blue}{-{x}^{5} \cdot \left(-1 \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right)} \]

   2. *-commutative 0.0%

      \[\leadsto -\color{blue}{\left(-1 \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot {x}^{5}} \]

   3. distribute-lft-neg-in 0.0%

      \[\leadsto \color{blue}{\left(-\left(-1 \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right)\right) \cdot {x}^{5}} \]

   4. unpow2 0.0%

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

   5. rem-square-sqrt 70.8%

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

   6. metadata-eval 70.8%

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

   7. metadata-eval 70.8%

      \[\leadsto \left(-\color{blue}{0}\right) \cdot {x}^{5} \]

   8. metadata-eval 70.8%

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

   9. mul0-lft 70.8%

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

7. Simplified 70.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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)))