x / (x^2 + 1)

Percentage Accurate: 77.1% → 100.0%

Time: 3.4s

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: 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + {x}^{-5}\right) - \left({x}^{-3} + {x}^{-7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2000000000.0)
   (/ 1.0 x)
   (if (<= x 100.0)
     (/ x (+ 1.0 (* x x)))
     (- (+ (/ 1.0 x) (pow x -5.0)) (+ (pow x -3.0) (pow x -7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -2000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 100.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = ((1.0 / x) + pow(x, -5.0)) - (pow(x, -3.0) + pow(x, -7.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2000000000.0d0)) then
        tmp = 1.0d0 / x
    else if (x <= 100.0d0) then
        tmp = x / (1.0d0 + (x * x))
    else
        tmp = ((1.0d0 / x) + (x ** (-5.0d0))) - ((x ** (-3.0d0)) + (x ** (-7.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 100.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = ((1.0 / x) + Math.pow(x, -5.0)) - (Math.pow(x, -3.0) + Math.pow(x, -7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2000000000.0:
		tmp = 1.0 / x
	elif x <= 100.0:
		tmp = x / (1.0 + (x * x))
	else:
		tmp = ((1.0 / x) + math.pow(x, -5.0)) - (math.pow(x, -3.0) + math.pow(x, -7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2000000000.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 100.0)
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) + (x ^ -5.0)) - Float64((x ^ -3.0) + (x ^ -7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2000000000.0)
		tmp = 1.0 / x;
	elseif (x <= 100.0)
		tmp = x / (1.0 + (x * x));
	else
		tmp = ((1.0 / x) + (x ^ -5.0)) - ((x ^ -3.0) + (x ^ -7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2000000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 100.0], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -3.0], $MachinePrecision] + N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e9

   1. Initial program 47.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2e9 < x < 100

   1. Initial program 100.0%

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

   if 100 < x

   1. Initial program 44.0%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)} \]
   3. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)\right)} \]

      2. pow-flip 100.0%

         \[\leadsto \color{blue}{{x}^{\left(-5\right)}} + \left(\frac{1}{x} - \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)\right) \]

      3. metadata-eval 100.0%

         \[\leadsto {x}^{\color{blue}{-5}} + \left(\frac{1}{x} - \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)\right) \]

      4. pow-flip 100.0%

         \[\leadsto {x}^{-5} + \left(\frac{1}{x} - \left(\color{blue}{{x}^{\left(-3\right)}} + \frac{1}{{x}^{7}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto {x}^{-5} + \left(\frac{1}{x} - \left({x}^{\color{blue}{-3}} + \frac{1}{{x}^{7}}\right)\right) \]

      6. pow-flip 100.0%

         \[\leadsto {x}^{-5} + \left(\frac{1}{x} - \left({x}^{-3} + \color{blue}{{x}^{\left(-7\right)}}\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-5} + \left(\frac{1}{x} - \left({x}^{-3} + {x}^{\color{blue}{-7}}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{x}^{-5} + \left(\frac{1}{x} - \left({x}^{-3} + {x}^{-7}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r- 100.0%

         \[\leadsto \color{blue}{\left({x}^{-5} + \frac{1}{x}\right) - \left({x}^{-3} + {x}^{-7}\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + {x}^{-5}\right)} - \left({x}^{-3} + {x}^{-7}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + {x}^{-5}\right) - \left({x}^{-3} + {x}^{-7}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + {x}^{-5}\right) - \left({x}^{-3} + {x}^{-7}\right)\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2000000000.0)
   (/ 1.0 x)
   (if (<= x 200000000.0) (/ x (+ 1.0 (* x x))) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -2000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 200000000.0) {
		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 <= (-2000000000.0d0)) then
        tmp = 1.0d0 / x
    else if (x <= 200000000.0d0) 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 <= -2000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 200000000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2000000000.0:
		tmp = 1.0 / x
	elif x <= 200000000.0:
		tmp = x / (1.0 + (x * x))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2000000000.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 200000000.0)
		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 <= -2000000000.0)
		tmp = 1.0 / x;
	elseif (x <= 200000000.0)
		tmp = x / (1.0 + (x * x));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2000000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 200000000.0], 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 < -2e9 or 2e8 < x

   1. Initial program 44.4%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2e9 < x < 2e8

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 98.9% 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 47.6%

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

   2. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.0%

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

Alternative 4: 52.0% 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 74.8%

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

2. Taylor expanded in x around 0 53.2%

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

3. Final simplification 53.2%

   \[\leadsto x \]

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 2023207 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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