x / (x^2 + 1)

Percentage Accurate: 76.0% → 100.0%

Time: 3.7s

Alternatives: 5

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

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5000.0)
    (/ x_m (+ 1.0 (* x_m x_m)))
    (- (/ 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 / (1.0 + (x_m * x_m));
	} 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 / (1.0d0 + (x_m * x_m))
    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 / (1.0 + (x_m * x_m));
	} 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 / (1.0 + (x_m * x_m))
	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(1.0 + Float64(x_m * x_m)));
	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 / (1.0 + (x_m * x_m));
	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[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $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 52.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 52.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]

      2. inv-pow 52.1%

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

      3. fma-define 52.1%

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

   4. Applied egg-rr 52.1%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. div-sub 100.0%

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

      2. associate-/r* 100.0%

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

      3. pow-plus 100.0%

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

      4. metadata-eval 100.0%

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

      5. exp-to-pow 100.0%

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

      6. *-commutative 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-lft-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-commutative 100.0%

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

      11. exp-to-pow 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 89.8%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot {\left(x\_m + \frac{1}{x\_m}\right)}^{-1} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (pow (+ x_m (/ 1.0 x_m)) -1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * pow((x_m + (1.0 / x_m)), -1.0);
}

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 + (1.0d0 / x_m)) ** (-1.0d0))
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 * Math.pow((x_m + (1.0 / x_m)), -1.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * math.pow((x_m + (1.0 / x_m)), -1.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * (Float64(x_m + Float64(1.0 / x_m)) ^ -1.0))
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 + (1.0 / x_m)) ^ -1.0);
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 * N[Power[N[(x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

   \[\frac{x}{x \cdot x + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 77.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]

   2. inv-pow 77.3%

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

   3. fma-define 77.3%

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

4. Applied egg-rr 77.3%

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

5. Taylor expanded in x around inf 77.3%

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

   1. distribute-rgt-in 77.3%

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

   2. *-lft-identity 77.3%

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

   3. unpow2 77.3%

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

   4. associate-/r* 77.4%

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

   5. *-rgt-identity 77.4%

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

   6. associate-*r/ 77.3%

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

   7. unpow-1 77.3%

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

   8. unpow-1 77.3%

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

   9. pow-sqr 77.4%

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

   10. metadata-eval 77.4%

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

   11. pow-plus 99.8%

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

   12. metadata-eval 99.8%

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

   13. unpow-1 99.8%

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

7. Simplified 99.8%

   \[\leadsto {\color{blue}{\left(x + \frac{1}{x}\right)}}^{-1} \]
8. Add Preprocessing

Alternative 3: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000000:\\ \;\;\;\;\frac{x\_m}{1 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 100000000.0) (/ x_m (+ 1.0 (* x_m 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 <= 100000000.0) {
		tmp = x_m / (1.0 + (x_m * 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 <= 100000000.0d0) then
        tmp = x_m / (1.0d0 + (x_m * 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 <= 100000000.0) {
		tmp = x_m / (1.0 + (x_m * 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 <= 100000000.0:
		tmp = x_m / (1.0 + (x_m * 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 <= 100000000.0)
		tmp = Float64(x_m / Float64(1.0 + Float64(x_m * 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 <= 100000000.0)
		tmp = x_m / (1.0 + (x_m * 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, 100000000.0], N[(x$95$m / N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e8

   1. Initial program 86.2%

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

   if 1e8 < x

   1. Initial program 52.0%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 100000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% 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 #s(literal 1 binary64) 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 68.4%

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

   if 1 < x

   1. Initial program 52.7%

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

   3. Taylor expanded in x around inf 99.3%

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

Alternative 5: 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 #s(literal 1 binary64) 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 77.4%

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

3. Taylor expanded in x around 0 51.5%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer Target 1: 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 2024137 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ x (/ 1 x))))

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