x / (x^2 + 1)

Percentage Accurate: 76.0% → 100.0%

Time: 2.3s

Alternatives: 4

Speedup: 1.0×

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1000000 \lor \neg \left(x \leq 5000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1000000.0) (not (<= x 5000.0)))
   (- (/ 1.0 x) (pow x -3.0))
   (/ x (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1000000.0) || !(x <= 5000.0)) {
		tmp = (1.0 / x) - pow(x, -3.0);
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1000000.0d0)) .or. (.not. (x <= 5000.0d0))) then
        tmp = (1.0d0 / x) - (x ** (-3.0d0))
    else
        tmp = x / (1.0d0 + (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1000000.0) || !(x <= 5000.0)) {
		tmp = (1.0 / x) - Math.pow(x, -3.0);
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1000000.0) or not (x <= 5000.0):
		tmp = (1.0 / x) - math.pow(x, -3.0)
	else:
		tmp = x / (1.0 + (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1000000.0) || !(x <= 5000.0))
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	else
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1000000.0) || ~((x <= 5000.0)))
		tmp = (1.0 / x) - (x ^ -3.0);
	else
		tmp = x / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1000000.0], N[Not[LessEqual[x, 5000.0]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e6 or 5e3 < x

   1. Initial program 56.0%

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

      1. clear-num 56.1%

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

      2. associate-/r/ 55.8%

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

      3. fma-def 55.8%

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

   3. Applied egg-rr 55.8%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow-1 100.0%

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

      2. exp-to-pow 48.3%

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

      3. *-commutative 48.3%

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

      4. log-pow 48.3%

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

      5. associate-*r* 48.3%

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

      6. metadata-eval 48.3%

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

      7. *-commutative 48.3%

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

      8. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   if -1e6 < x < 5e3

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000 \lor \neg \left(x \leq 5000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e+20)
   (/ 1.0 x)
   (if (<= x 100000000.0) (/ x (+ 1.0 (* x x))) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e+20) {
		tmp = 1.0 / x;
	} else if (x <= 100000000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d+20)) then
        tmp = 1.0d0 / x
    else if (x <= 100000000.0d0) then
        tmp = x / (1.0d0 + (x * x))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e+20) {
		tmp = 1.0 / x;
	} else if (x <= 100000000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e+20:
		tmp = 1.0 / x
	elif x <= 100000000.0:
		tmp = x / (1.0 + (x * x))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e+20)
		tmp = Float64(1.0 / x);
	elseif (x <= 100000000.0)
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e+20)
		tmp = 1.0 / x;
	elseif (x <= 100000000.0)
		tmp = x / (1.0 + (x * x));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e+20], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e20 or 1e8 < x

   1. Initial program 53.6%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -5e20 < x < 1e8

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ 1.0 x) (if (<= x 1.0) x (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / x;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0 / x
    else if (x <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / x;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0 / x
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0 / x;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.0], x, N[(1.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 56.7%

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

   2. Taylor expanded in x around inf 98.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4: 50.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 80.0%

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

2. Taylor expanded in x around 0 55.2%

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

3. Final simplification 55.2%

   \[\leadsto x \]

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 2023215 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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