x / (x^2 + 1)

Percentage Accurate: 77.3% → 100.0%

Time: 5.3s

Alternatives: 3

Speedup: N/A×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 77.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\left(x\_m \cdot x\_m\right)}^{-1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2000:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left({\left(\left(x\_m \cdot x\_m\right) \cdot -1\right)}^{2}, 1, -1\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, 1, -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({t\_0}^{2}, 1, 1\right)}{x\_m} - \frac{\mathsf{fma}\left(t\_0, 1, {\left({\left(x\_m \cdot x\_m\right)}^{3}\right)}^{-1}\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (pow (* x_m x_m) -1.0)))
   (*
    x_s
    (if (<= x_m 2000.0)
      (/
       x_m
       (/
        (fma (pow (* (* x_m x_m) -1.0) 2.0) 1.0 -1.0)
        (fma (* x_m x_m) 1.0 -1.0)))
      (-
       (/ (fma (pow t_0 2.0) 1.0 1.0) x_m)
       (/ (fma t_0 1.0 (pow (pow (* x_m x_m) 3.0) -1.0)) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = pow((x_m * x_m), -1.0);
	double tmp;
	if (x_m <= 2000.0) {
		tmp = x_m / (fma(pow(((x_m * x_m) * -1.0), 2.0), 1.0, -1.0) / fma((x_m * x_m), 1.0, -1.0));
	} else {
		tmp = (fma(pow(t_0, 2.0), 1.0, 1.0) / x_m) - (fma(t_0, 1.0, pow(pow((x_m * x_m), 3.0), -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)
	t_0 = Float64(x_m * x_m) ^ -1.0
	tmp = 0.0
	if (x_m <= 2000.0)
		tmp = Float64(x_m / Float64(fma((Float64(Float64(x_m * x_m) * -1.0) ^ 2.0), 1.0, -1.0) / fma(Float64(x_m * x_m), 1.0, -1.0)));
	else
		tmp = Float64(Float64(fma((t_0 ^ 2.0), 1.0, 1.0) / x_m) - Float64(fma(t_0, 1.0, ((Float64(x_m * x_m) ^ 3.0) ^ -1.0)) / 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_] := Block[{t$95$0 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2000.0], N[(x$95$m / N[(N[(N[Power[N[(N[(x$95$m * x$95$m), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] - N[(N[(t$95$0 * 1.0 + N[Power[N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 3.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e3

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   if 2e3 < x

   1. Initial program 53.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{{x}^{4}}\right) - \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}\right)}{x}} \]
   3. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{1 + \frac{1}{{x}^{4}}}{x} - \color{blue}{\frac{\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}}{x}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1 + \frac{1}{{x}^{4}}}{x} - \color{blue}{\frac{\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}}{x}} \]

   4. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\left(x\_m \cdot x\_m\right)}^{-1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100:\\ \;\;\;\;\frac{x\_m}{x\_m \cdot x\_m + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({t\_0}^{2}, 1, 1\right)}{x\_m} - \frac{\mathsf{fma}\left(t\_0, 1, {\left({\left(x\_m \cdot x\_m\right)}^{3}\right)}^{-1}\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (pow (* x_m x_m) -1.0)))
   (*
    x_s
    (if (<= x_m 100.0)
      (/ x_m (+ (* x_m x_m) 1.0))
      (-
       (/ (fma (pow t_0 2.0) 1.0 1.0) x_m)
       (/ (fma t_0 1.0 (pow (pow (* x_m x_m) 3.0) -1.0)) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = pow((x_m * x_m), -1.0);
	double tmp;
	if (x_m <= 100.0) {
		tmp = x_m / ((x_m * x_m) + 1.0);
	} else {
		tmp = (fma(pow(t_0, 2.0), 1.0, 1.0) / x_m) - (fma(t_0, 1.0, pow(pow((x_m * x_m), 3.0), -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)
	t_0 = Float64(x_m * x_m) ^ -1.0
	tmp = 0.0
	if (x_m <= 100.0)
		tmp = Float64(x_m / Float64(Float64(x_m * x_m) + 1.0));
	else
		tmp = Float64(Float64(fma((t_0 ^ 2.0), 1.0, 1.0) / x_m) - Float64(fma(t_0, 1.0, ((Float64(x_m * x_m) ^ 3.0) ^ -1.0)) / 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_] := Block[{t$95$0 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 100.0], N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] - N[(N[(t$95$0 * 1.0 + N[Power[N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 3.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 100

   1. Initial program 100.0%

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

   if 100 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{{x}^{4}}\right) - \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}\right)}{x}} \]
   3. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{1 + \frac{1}{{x}^{4}}}{x} - \color{blue}{\frac{\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}}{x}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1 + \frac{1}{{x}^{4}}}{x} - \color{blue}{\frac{\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}}{x}} \]

   4. Applied rewrites 100.0%

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

Alternative 3: 49.5% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\left(x\_m \cdot x\_m\right)}^{-1}\\ x\_s \cdot \left(\frac{\mathsf{fma}\left({t\_0}^{2}, 1, 1\right)}{x\_m} - \frac{\mathsf{fma}\left(t\_0, 1, {\left({\left(x\_m \cdot x\_m\right)}^{3}\right)}^{-1}\right)}{x\_m}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (pow (* x_m x_m) -1.0)))
   (*
    x_s
    (-
     (/ (fma (pow t_0 2.0) 1.0 1.0) x_m)
     (/ (fma t_0 1.0 (pow (pow (* x_m x_m) 3.0) -1.0)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = pow((x_m * x_m), -1.0);
	return x_s * ((fma(pow(t_0, 2.0), 1.0, 1.0) / x_m) - (fma(t_0, 1.0, pow(pow((x_m * x_m), 3.0), -1.0)) / x_m));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * x_m) ^ -1.0
	return Float64(x_s * Float64(Float64(fma((t_0 ^ 2.0), 1.0, 1.0) / x_m) - Float64(fma(t_0, 1.0, ((Float64(x_m * x_m) ^ 3.0) ^ -1.0)) / 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_] := Block[{t$95$0 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]}, N[(x$95$s * N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] - N[(N[(t$95$0 * 1.0 + N[Power[N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 3.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.3%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{{x}^{4}}\right) - \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}\right)}{x}} \]
3. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \frac{1 + \frac{1}{{x}^{4}}}{x} - \color{blue}{\frac{\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}}{x}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{{x}^{4}}}{x} - \color{blue}{\frac{\frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}}{x}} \]

4. Applied rewrites 49.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{2}, 1, 1\right)}{x} - \frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{-1}, 1, {\left({\left(x \cdot x\right)}^{3}\right)}^{-1}\right)}{x}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64
  (/ x (+ (* x x) 1.0)))