x / (x^2 + 1)

Percentage Accurate: 76.0% → 100.0%

Time: 2.3s

Alternatives: 4

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * x) + 1.0d0)
end function

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * x) + 1.0d0)
end function

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\frac{x\_m}{x\_m \cdot x\_m + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} - {x\_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5000.0)
    (/ x_m (+ (* x_m x_m) 1.0))
    (- (/ 1.0 x_m) (pow x_m -3.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5000.0) {
		tmp = x_m / ((x_m * x_m) + 1.0);
	} else {
		tmp = (1.0 / x_m) - pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5000.0d0) then
        tmp = x_m / ((x_m * x_m) + 1.0d0)
    else
        tmp = (1.0d0 / x_m) - (x_m ** (-3.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5000.0) {
		tmp = x_m / ((x_m * x_m) + 1.0);
	} else {
		tmp = (1.0 / x_m) - Math.pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5000.0:
		tmp = x_m / ((x_m * x_m) + 1.0)
	else:
		tmp = (1.0 / x_m) - math.pow(x_m, -3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(x_m / Float64(Float64(x_m * x_m) + 1.0));
	else
		tmp = Float64(Float64(1.0 / x_m) - (x_m ^ -3.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5000.0)
		tmp = x_m / ((x_m * x_m) + 1.0);
	else
		tmp = (1.0 / x_m) - (x_m ^ -3.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] - N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 86.2%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   if 5e3 < x

   1. Initial program 58.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. inv-pow 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{{\left({x}^{3}\right)}^{-1}} \]

      2. cube-mult 100.0%

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

      3. add-cube-cbrt 100.0%

         \[\leadsto \frac{1}{x} - {\left(\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(x \cdot x\right)\right)}^{-1} \]

      4. associate-*l* 100.0%

         \[\leadsto \frac{1}{x} - {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(x \cdot x\right)\right)\right)}}^{-1} \]

      5. unpow-prod-down 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x} \cdot \left(x \cdot x\right)\right)}^{-1}} \]

      6. pow2 100.0%

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

      7. pow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{x} - \color{blue}{{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1} \cdot {\left(\sqrt[3]{x} \cdot {x}^{2}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. unpow-1 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-/r* 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{\frac{\frac{{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1}}{\sqrt[3]{x}}}{{x}^{2}}} \]

      6. unpow-1 100.0%

         \[\leadsto \frac{1}{x} - \frac{\frac{\color{blue}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{2}}}}{\sqrt[3]{x}}}{{x}^{2}} \]

      7. associate-/r* 100.0%

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

      8. unpow2 100.0%

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

      9. rem-3cbrt-lft 100.0%

         \[\leadsto \frac{1}{x} - \frac{\frac{1}{\color{blue}{x}}}{{x}^{2}} \]

      10. associate-/r* 100.0%

          \[\leadsto \frac{1}{x} - \color{blue}{\frac{1}{x \cdot {x}^{2}}} \]

      11. unpow2 100.0%

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

      12. cube-mult 100.0%

          \[\leadsto \frac{1}{x} - \frac{1}{\color{blue}{{x}^{3}}} \]

      13. rem-exp-log 100.0%

          \[\leadsto \frac{1}{x} - \frac{1}{\color{blue}{e^{\log \left({x}^{3}\right)}}} \]

      14. rec-exp 100.0%

          \[\leadsto \frac{1}{x} - \color{blue}{e^{-\log \left({x}^{3}\right)}} \]

      15. log-pow 100.0%

          \[\leadsto \frac{1}{x} - e^{-\color{blue}{3 \cdot \log x}} \]

      16. distribute-lft-neg-in 100.0%

          \[\leadsto \frac{1}{x} - e^{\color{blue}{\left(-3\right) \cdot \log x}} \]

      17. metadata-eval 100.0%

          \[\leadsto \frac{1}{x} - e^{\color{blue}{-3} \cdot \log x} \]

      18. *-commutative 100.0%

          \[\leadsto \frac{1}{x} - e^{\color{blue}{\log x \cdot -3}} \]

      19. exp-to-pow 100.0%

          \[\leadsto \frac{1}{x} - \color{blue}{{x}^{-3}} \]

   7. Simplified 100.0%

      \[\leadsto \frac{1}{x} - \color{blue}{{x}^{-3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 200000000:\\ \;\;\;\;\frac{x\_m}{x\_m \cdot x\_m + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 200000000.0) (/ x_m (+ (* x_m x_m) 1.0)) (/ 1.0 x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 200000000.0) {
		tmp = x_m / ((x_m * x_m) + 1.0);
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 200000000.0d0) then
        tmp = x_m / ((x_m * x_m) + 1.0d0)
    else
        tmp = 1.0d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 200000000.0) {
		tmp = x_m / ((x_m * x_m) + 1.0);
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 200000000.0:
		tmp = x_m / ((x_m * x_m) + 1.0)
	else:
		tmp = 1.0 / x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 200000000.0)
		tmp = Float64(x_m / Float64(Float64(x_m * x_m) + 1.0));
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 200000000.0)
		tmp = x_m / ((x_m * x_m) + 1.0);
	else
		tmp = 1.0 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 200000000.0], N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e8

   1. Initial program 86.3%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   if 2e8 < x

   1. Initial program 57.4%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200000000:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.0) x_m (/ 1.0 x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = x_m
    else
        tmp = 1.0d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = x_m
	else:
		tmp = 1.0 / x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], x$95$m, N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.2%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 67.6%

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

   if 1 < x

   1. Initial program 58.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 78.4%

   \[\frac{x}{x \cdot x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 50.0%

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

4. Final simplification 50.0%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x (/ 1.0 x))))

C

double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Python

def code(x):
	return 1.0 / (x + (1.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(x + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x + (1.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024039 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))