x / (x^2 + 1)

Percentage Accurate: 77.1% → 100.0%

Time: 3.6s

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: 77.1% 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, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 500:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) + -1} \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

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 500.0)
    (*
     (/ 1.0 (+ (* x_m (* x_m (* x_m x_m))) -1.0))
     (* x_m (+ (* x_m x_m) -1.0)))
    (/ (+ 1.0 (/ -1.0 (* x_m x_m))) 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 <= 500.0) {
		tmp = (1.0 / ((x_m * (x_m * (x_m * x_m))) + -1.0)) * (x_m * ((x_m * x_m) + -1.0));
	} else {
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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 <= 500.0d0) then
        tmp = (1.0d0 / ((x_m * (x_m * (x_m * x_m))) + (-1.0d0))) * (x_m * ((x_m * x_m) + (-1.0d0)))
    else
        tmp = (1.0d0 + ((-1.0d0) / (x_m * x_m))) / 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 <= 500.0) {
		tmp = (1.0 / ((x_m * (x_m * (x_m * x_m))) + -1.0)) * (x_m * ((x_m * x_m) + -1.0));
	} else {
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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 <= 500.0:
		tmp = (1.0 / ((x_m * (x_m * (x_m * x_m))) + -1.0)) * (x_m * ((x_m * x_m) + -1.0))
	else:
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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 <= 500.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * Float64(x_m * Float64(x_m * x_m))) + -1.0)) * Float64(x_m * Float64(Float64(x_m * x_m) + -1.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(x_m * x_m))) / 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 <= 500.0)
		tmp = (1.0 / ((x_m * (x_m * (x_m * x_m))) + -1.0)) * (x_m * ((x_m * x_m) + -1.0));
	else
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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, 500.0], N[(N[(1.0 / N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 86.1%

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

      1. clear-num N/A

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

      2. flip-+ N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r/ N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. neg-mul-1 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \left(x \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      10. fma-define N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \mathsf{fma}\left(x, x \cdot x, \mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \mathsf{fma}\left(x, x \cdot x, \mathsf{neg}\left(x\right)\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \mathsf{fma}\left(x, x \cdot x, \mathsf{neg}\left(x \cdot 1\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \mathsf{fma}\left(x, x \cdot x, \mathsf{neg}\left(1 \cdot x\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}\right), \color{blue}{\left(\mathsf{fma}\left(x, x \cdot x, \mathsf{neg}\left(1 \cdot x\right)\right)\right)}\right) \]

   4. Applied egg-rr 73.5%

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

   if 500 < x

   1. Initial program 57.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - \frac{1}{{x}^{2}}\right), \color{blue}{x}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{{x}^{2}}\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{{x}^{2}}\right)\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left({x}^{2}\right)\right)\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(x \cdot x\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   5. Simplified 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 500:\\ \;\;\;\;\frac{x\_m}{1 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

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 500.0)
    (/ x_m (+ 1.0 (* x_m x_m)))
    (/ (+ 1.0 (/ -1.0 (* x_m x_m))) 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 <= 500.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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 <= 500.0d0) then
        tmp = x_m / (1.0d0 + (x_m * x_m))
    else
        tmp = (1.0d0 + ((-1.0d0) / (x_m * x_m))) / 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 <= 500.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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 <= 500.0:
		tmp = x_m / (1.0 + (x_m * x_m))
	else:
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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 <= 500.0)
		tmp = Float64(x_m / Float64(1.0 + Float64(x_m * x_m)));
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(x_m * x_m))) / 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 <= 500.0)
		tmp = x_m / (1.0 + (x_m * x_m));
	else
		tmp = (1.0 + (-1.0 / (x_m * x_m))) / 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, 500.0], N[(x$95$m / N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 86.1%

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

   if 500 < x

   1. Initial program 57.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - \frac{1}{{x}^{2}}\right), \color{blue}{x}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{{x}^{2}}\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{{x}^{2}}\right)\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left({x}^{2}\right)\right)\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(x \cdot x\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   5. Simplified 100.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{x \cdot x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 50000000:\\ \;\;\;\;\frac{x\_m}{1 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

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 50000000.0) (/ 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 <= 50000000.0) {
		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 <= 50000000.0d0) 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 <= 50000000.0) {
		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 <= 50000000.0:
		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 <= 50000000.0)
		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 <= 50000000.0)
		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, 50000000.0], 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 < 5e7

   1. Initial program 86.1%

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

   if 5e7 < x

   1. Initial program 57.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 100.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% 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 86.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 64.2%

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

   if 1 < x

   1. Initial program 58.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 98.6%

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

Alternative 5: 51.4% 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 80.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 51.5%

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

Developer Target 1: 99.8% 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 2024163 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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