x / (x^2 + 1)

Percentage Accurate: 76.5% → 100.0%

Time: 2.1s

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: 76.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10000 \lor \neg \left(x \leq 5000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -10000.0) (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 <= -10000.0) || !(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 <= (-10000.0d0)) .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 <= -10000.0) || !(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 <= -10000.0) 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 <= -10000.0) || !(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 <= -10000.0) || ~((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, -10000.0], 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 < -1e4 or 5e3 < x

   1. Initial program 54.5%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. add-log-exp 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. log-prod 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto \frac{1}{x} - \left(\color{blue}{0} + \log \left(e^{\frac{1}{{x}^{3}}}\right)\right) \]

      5. add-log-exp 100.0%

         \[\leadsto \frac{1}{x} - \left(0 + \color{blue}{\frac{1}{{x}^{3}}}\right) \]

      6. pow-flip 100.0%

         \[\leadsto \frac{1}{x} - \left(0 + \color{blue}{{x}^{\left(-3\right)}}\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1}{x} - \left(0 + {x}^{\color{blue}{-3}}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{x} - \color{blue}{\left(0 + {x}^{-3}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   if -1e4 < 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000 \lor \neg \left(x \leq 5000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+77} \lor \neg \left(x \leq 400000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2e+77) (not (<= x 400000.0))) (/ 1.0 x) (/ x (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -2e+77) || !(x <= 400000.0)) {
		tmp = 1.0 / x;
	} 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 <= (-2d+77)) .or. (.not. (x <= 400000.0d0))) then
        tmp = 1.0d0 / x
    else
        tmp = x / (1.0d0 + (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2e+77) || !(x <= 400000.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2e+77) or not (x <= 400000.0):
		tmp = 1.0 / x
	else:
		tmp = x / (1.0 + (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2e+77) || !(x <= 400000.0))
		tmp = Float64(1.0 / x);
	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 <= -2e+77) || ~((x <= 400000.0)))
		tmp = 1.0 / x;
	else
		tmp = x / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2e+77], N[Not[LessEqual[x, 400000.0]], $MachinePrecision]], N[(1.0 / x), $MachinePrecision], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999997e77 or 4e5 < x

   1. Initial program 50.3%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -1.99999999999999997e77 < x < 4e5

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+77} \lor \neg \left(x \leq 400000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ 1.0 x) x))

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = 1.0 / x
	else:
		tmp = x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 / x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 54.5%

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

   2. Taylor expanded in x around inf 99.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.4%

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

Alternative 4: 51.6% 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 77.1%

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

2. Taylor expanded in x around 0 51.2%

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

3. Final simplification 51.2%

   \[\leadsto x \]

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 2023309 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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