x / (x^2 + 1)

Percentage Accurate: 76.6% → 99.9%

Time: 3.1s

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: 76.6% 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: 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]

Derivation

1. Initial program 76.2%

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

   1. clear-num 76.2%

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

   2. inv-pow 76.2%

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

   3. fma-def 76.2%

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

3. Applied egg-rr 76.2%

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

4. Taylor expanded in x around 0 99.8%

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

   1. add-log-exp 8.1%

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

   2. *-un-lft-identity 8.1%

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

   3. log-prod 8.1%

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

   4. metadata-eval 8.1%

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

   5. add-log-exp 99.8%

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

   6. unpow-1 99.8%

      \[\leadsto 0 + \color{blue}{\frac{1}{x + \frac{1}{x}}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{0 + \frac{1}{x + \frac{1}{x}}} \]
7. Step-by-step derivation

   1. +-lft-identity 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.85) (/ 1.0 x) (if (<= x 0.86) (* x (- 1.0 (* x x))) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.85) {
		tmp = 1.0 / x;
	} else if (x <= 0.86) {
		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 <= (-0.85d0)) then
        tmp = 1.0d0 / x
    else if (x <= 0.86d0) 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 <= -0.85) {
		tmp = 1.0 / x;
	} else if (x <= 0.86) {
		tmp = x * (1.0 - (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.85:
		tmp = 1.0 / x
	elif x <= 0.86:
		tmp = x * (1.0 - (x * x))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.85)
		tmp = Float64(1.0 / x);
	elseif (x <= 0.86)
		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 <= -0.85)
		tmp = 1.0 / x;
	elseif (x <= 0.86)
		tmp = x * (1.0 - (x * x));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.85], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 0.86], 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 < -0.849999999999999978 or 0.859999999999999987 < x

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf 99.4%

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

   if -0.849999999999999978 < x < 0.859999999999999987

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. unsub-neg 98.9%

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

   4. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. unpow3 98.9%

         \[\leadsto 1 \cdot x - \color{blue}{\left(x \cdot x\right) \cdot x} \]

      3. distribute-rgt-out-- 98.9%

         \[\leadsto \color{blue}{x \cdot \left(1 - x \cdot x\right)} \]

   6. Applied egg-rr 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

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 53.3%

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

   2. Taylor expanded in x around inf 99.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.8%

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

Alternative 4: 51.3% 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 76.2%

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

2. Taylor expanded in x around 0 50.2%

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

3. Final simplification 50.2%

   \[\leadsto x \]

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 2023293 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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