x / (x^2 + 1)

Percentage Accurate: 76.7% → 100.0%

Time: 5.4s

Alternatives: 6

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: 76.7% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x\_m - {x\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(1 + \frac{1}{{x\_m}^{2}}\right)}\\ \end{array} \end{array} \]

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 5e-8)
    (- x_m (pow x_m 3.0))
    (/ 1.0 (* x_m (+ 1.0 (/ 1.0 (pow x_m 2.0))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5e-8) {
		tmp = x_m - pow(x_m, 3.0);
	} else {
		tmp = 1.0 / (x_m * (1.0 + (1.0 / pow(x_m, 2.0))));
	}
	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 <= 5d-8) then
        tmp = x_m - (x_m ** 3.0d0)
    else
        tmp = 1.0d0 / (x_m * (1.0d0 + (1.0d0 / (x_m ** 2.0d0))))
    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 <= 5e-8) {
		tmp = x_m - Math.pow(x_m, 3.0);
	} else {
		tmp = 1.0 / (x_m * (1.0 + (1.0 / Math.pow(x_m, 2.0))));
	}
	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 <= 5e-8:
		tmp = x_m - math.pow(x_m, 3.0)
	else:
		tmp = 1.0 / (x_m * (1.0 + (1.0 / math.pow(x_m, 2.0))))
	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 <= 5e-8)
		tmp = Float64(x_m - (x_m ^ 3.0));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(1.0 + Float64(1.0 / (x_m ^ 2.0)))));
	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 <= 5e-8)
		tmp = x_m - (x_m ^ 3.0);
	else
		tmp = 1.0 / (x_m * (1.0 + (1.0 / (x_m ^ 2.0))));
	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, 5e-8], N[(x$95$m - N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(1.0 + N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999998e-8

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0 67.7%

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

      1. mul-1-neg 67.7%

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

      2. distribute-rgt-in 67.8%

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

      3. *-lft-identity 67.8%

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

      4. distribute-lft-neg-out 67.8%

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

      5. unpow2 67.8%

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

      6. unpow3 67.8%

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

      7. unsub-neg 67.8%

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

   5. Simplified 67.8%

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

   if 4.9999999999999998e-8 < x

   1. Initial program 63.8%

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

      1. frac-2neg 63.8%

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

      2. div-inv 63.7%

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

      3. fma-define 63.7%

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

   4. Applied egg-rr 63.7%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{-\mathsf{fma}\left(x, x, 1\right)}} \]
   5. Step-by-step derivation

      1. un-div-inv 63.8%

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

      2. clear-num 63.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 4.4%

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

      5. sqr-neg 4.4%

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

      6. sqrt-unprod 4.1%

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

      7. add-sqr-sqrt 4.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 61.2%

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

      10. sqr-neg 61.2%

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

      11. sqrt-unprod 63.7%

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

      12. add-sqr-sqrt 63.9%

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

   6. Applied egg-rr 63.9%

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

   7. Taylor expanded in x around inf 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

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 5000.0)
    (* x_m (/ 1.0 (fma 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 <= 5000.0) {
		tmp = x_m * (1.0 / fma(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 <= 5000.0)
		tmp = Float64(x_m * Float64(1.0 / fma(x_m, x_m, 1.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 / x_m) / x_m)) / x_m);
	end
	return Float64(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 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 83.7%

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

      1. frac-2neg 83.7%

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

      2. div-inv 83.7%

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

      3. fma-define 83.7%

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

   4. Applied egg-rr 83.7%

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

   if 5e3 < x

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. metadata-eval 100.0%

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

      2. unpow2 100.0%

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

      3. frac-times 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{-1}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

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 5000.0)
    (/ 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 <= 5000.0) {
		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 <= 5000.0d0) 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 <= 5000.0) {
		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 <= 5000.0:
		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 <= 5000.0)
		tmp = Float64(x_m / Float64(1.0 + Float64(x_m * x_m)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 / 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 <= 5000.0)
		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, 5000.0], N[(x$95$m / N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 83.7%

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

   if 5e3 < x

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. metadata-eval 100.0%

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

      2. unpow2 100.0%

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

      3. frac-times 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 88.3%

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

Alternative 4: 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 84.0%

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

   if 5e7 < x

   1. Initial program 60.8%

      \[\frac{x}{x \cdot x + 1} \]
   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 88.3%

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

Alternative 5: 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 #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 83.7%

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

   3. Taylor expanded in x around 0 68.2%

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

   if 1 < x

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 98.9%

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

Alternative 6: 51.3% 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 77.8%

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

3. Taylor expanded in x around 0 50.0%

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

Developer target: 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 2024111 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :alt
  (/ 1.0 (+ x (/ 1.0 x)))

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