x / (x^2 + 1)

Percentage Accurate: 76.3% → 100.0%

Time: 2.5s

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: 76.3% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000000 \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -20000000000.0) (not (<= x 50000000.0)))
   (/ 1.0 x)
   (/ x (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -20000000000.0) || !(x <= 50000000.0)) {
		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 <= (-20000000000.0d0)) .or. (.not. (x <= 50000000.0d0))) 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 <= -20000000000.0) || !(x <= 50000000.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -20000000000.0) or not (x <= 50000000.0):
		tmp = 1.0 / x
	else:
		tmp = x / (1.0 + (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -20000000000.0) || !(x <= 50000000.0))
		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 <= -20000000000.0) || ~((x <= 50000000.0)))
		tmp = 1.0 / x;
	else
		tmp = x / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -20000000000.0], N[Not[LessEqual[x, 50000000.0]], $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 < -2e10 or 5e7 < x

   1. Initial program 49.8%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2e10 < x < 5e7

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000000 \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ x (hypot 1.0 x)) (hypot 1.0 x)))

C

double code(double x) {
	return (x / hypot(1.0, x)) / hypot(1.0, x);
}

Java

public static double code(double x) {
	return (x / Math.hypot(1.0, x)) / Math.hypot(1.0, x);
}

Python

def code(x):
	return (x / math.hypot(1.0, x)) / math.hypot(1.0, x)

Julia

function code(x)
	return Float64(Float64(x / hypot(1.0, x)) / hypot(1.0, x))
end

MATLAB

function tmp = code(x)
	tmp = (x / hypot(1.0, x)) / hypot(1.0, x);
end

Wolfram

code[x_] := N[(N[(x / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

   1. add-sqr-sqrt 74.6%

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

   2. *-un-lft-identity 74.6%

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

   3. times-frac 74.8%

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

   4. +-commutative 74.8%

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

   5. hypot-1-def 74.8%

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

   6. +-commutative 74.8%

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

   7. hypot-1-def 100.0%

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

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)}} \]
4. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. *-un-lft-identity 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \frac{\frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \]

Alternative 3: 99.0% 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 50.5%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.3%

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

4. Final simplification 99.1%

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

Alternative 4: 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 74.7%

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

2. Taylor expanded in x around 0 49.9%

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

3. Final simplification 49.9%

   \[\leadsto x \]

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 2023320 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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