x / (x^2 + 1)

Percentage Accurate: 76.0% → 100.0%

Time: 3.2s

Alternatives: 5

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

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -10000.0) (not (<= x 10000.0)))
   (- (/ 1.0 x) (pow x -3.0))
   (* x (/ 1.0 (fma x x 1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -10000.0) || !(x <= 10000.0)) {
		tmp = (1.0 / x) - pow(x, -3.0);
	} else {
		tmp = x * (1.0 / fma(x, x, 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -10000.0) || !(x <= 10000.0))
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	else
		tmp = Float64(x * Float64(1.0 / fma(x, x, 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -10000.0], N[Not[LessEqual[x, 10000.0]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e4 or 1e4 < x

   1. Initial program 51.4%

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

      1. clear-num 51.5%

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

      2. associate-/r/ 51.3%

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

      3. fma-def 51.3%

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

   3. Applied egg-rr 51.3%

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

   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. exp-to-pow 53.1%

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

      2. *-commutative 53.1%

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

      3. exp-neg 53.1%

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

      4. distribute-lft-neg-in 53.1%

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

      5. metadata-eval 53.1%

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

      6. *-commutative 53.1%

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

      7. 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 -1e4 < x < 1e4

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

      3. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -12000 \lor \neg \left(x \leq 10000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -12000.0) (not (<= x 10000.0)))
   (- (/ 1.0 x) (pow x -3.0))
   (/ x (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -12000.0) || !(x <= 10000.0)) {
		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 <= (-12000.0d0)) .or. (.not. (x <= 10000.0d0))) 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 <= -12000.0) || !(x <= 10000.0)) {
		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 <= -12000.0) or not (x <= 10000.0):
		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 <= -12000.0) || !(x <= 10000.0))
		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 <= -12000.0) || ~((x <= 10000.0)))
		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, -12000.0], N[Not[LessEqual[x, 10000.0]], $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 < -12000 or 1e4 < x

   1. Initial program 51.4%

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

      1. clear-num 51.5%

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

      2. associate-/r/ 51.3%

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

      3. fma-def 51.3%

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

   3. Applied egg-rr 51.3%

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

   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. exp-to-pow 53.1%

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

      2. *-commutative 53.1%

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

      3. exp-neg 53.1%

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

      4. distribute-lft-neg-in 53.1%

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

      5. metadata-eval 53.1%

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

      6. *-commutative 53.1%

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

      7. 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 -12000 < x < 1e4

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12000 \lor \neg \left(x \leq 10000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e+24)
   (/ 1.0 x)
   (if (<= x 5000000.0) (/ x (+ 1.0 (* x x))) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e+24) {
		tmp = 1.0 / x;
	} else if (x <= 5000000.0) {
		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+24)) then
        tmp = 1.0d0 / x
    else if (x <= 5000000.0d0) 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+24) {
		tmp = 1.0 / x;
	} else if (x <= 5000000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e+24:
		tmp = 1.0 / x
	elif x <= 5000000.0:
		tmp = x / (1.0 + (x * x))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e+24)
		tmp = Float64(1.0 / x);
	elseif (x <= 5000000.0)
		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+24)
		tmp = 1.0 / x;
	elseif (x <= 5000000.0)
		tmp = x / (1.0 + (x * x));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e+24], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 5000000.0], 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 < -5.00000000000000045e24 or 5e6 < x

   1. Initial program 49.4%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -5.00000000000000045e24 < x < 5e6

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4: 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 52.5%

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

   2. Taylor expanded in x around inf 97.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

4. Final simplification 97.9%

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

Alternative 5: 51.2% 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 75.3%

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

2. Taylor expanded in x around 0 49.1%

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

3. Final simplification 49.1%

   \[\leadsto x \]

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 2023230 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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