x / (x^2 + 1)

Percentage Accurate: 76.7% → 99.8%

Time: 2.0s

Alternatives: 3

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: 76.7% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{-1}{x} - x} \end{array} \]

FPCore

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

C

double code(double x) {
	return -1.0 / ((-1.0 / x) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (((-1.0d0) / x) - x)
end function

Java

public static double code(double x) {
	return -1.0 / ((-1.0 / x) - x);
}

Python

def code(x):
	return -1.0 / ((-1.0 / x) - x)

Julia

function code(x)
	return Float64(-1.0 / Float64(Float64(-1.0 / x) - x))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / ((-1.0 / x) - x);
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

   1. frac-2neg 74.7%

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

   2. distribute-frac-neg 74.7%

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

   3. neg-sub0 74.7%

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

   4. clear-num 74.4%

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

   5. +-commutative 74.4%

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

   6. distribute-neg-in 74.4%

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

   7. unsub-neg 74.4%

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

   8. div-sub 74.4%

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

   9. metadata-eval 74.4%

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

   10. remove-double-neg 74.4%

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

   11. distribute-rgt-neg-in 74.4%

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

   12. distribute-rgt-neg-in 74.4%

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

   13. distribute-neg-frac 74.4%

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

   14. neg-sub0 74.4%

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

   15. metadata-eval 74.4%

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

   16. remove-double-neg 74.4%

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

   17. neg-sub0 74.4%

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

   18. sub-neg 74.4%

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

   19. distribute-neg-in 74.4%

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

   20. metadata-eval 74.4%

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

   21. remove-double-neg 74.4%

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

   22. flip-- 99.6%

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

   23. neg-sub0 99.6%

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

   24. remove-double-neg 99.6%

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

3. Applied egg-rr 99.6%

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

   1. sub0-neg 99.6%

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

   2. distribute-neg-frac 99.6%

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

   3. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.0% 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 49.5%

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

   2. Taylor expanded in x around inf 99.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.7%

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

Alternative 3: 51.1% 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 51.1%

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

3. Final simplification 51.1%

   \[\leadsto x \]

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 2023297 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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