Asymptote C

Percentage Accurate: 54.1% → 100.0%

Time: 8.4s

Alternatives: 7

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x} + 3}{\frac{1}{x} - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ (/ 1.0 x) 3.0) (- (/ 1.0 x) x)))

C

double code(double x) {
	return ((1.0 / x) + 3.0) / ((1.0 / x) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / x) + 3.0d0) / ((1.0d0 / x) - x)
end function

Java

public static double code(double x) {
	return ((1.0 / x) + 3.0) / ((1.0 / x) - x);
}

Python

def code(x):
	return ((1.0 / x) + 3.0) / ((1.0 / x) - x)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / x) + 3.0) / Float64(Float64(1.0 / x) - x))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / x) + 3.0) / ((1.0 / x) - x);
end

Wolfram

code[x_] := N[(N[(N[(1.0 / x), $MachinePrecision] + 3.0), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

   1. clear-num 52.6%

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

   2. frac-2neg 52.6%

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

   3. frac-sub 52.6%

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

   4. *-un-lft-identity 52.6%

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

   5. neg-sub0 52.6%

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

   6. associate-+l- 52.6%

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

   7. neg-sub0 52.6%

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

   8. +-commutative 52.6%

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

   9. sub-neg 52.6%

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

   10. neg-sub0 52.6%

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

   11. +-commutative 52.6%

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

   12. associate--r+ 52.6%

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

   13. metadata-eval 52.6%

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

   14. neg-sub0 52.6%

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

   15. associate-+l- 52.6%

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

   16. neg-sub0 52.6%

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

   17. +-commutative 52.6%

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

   18. sub-neg 52.6%

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

3. Applied egg-rr 52.6%

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

4. Taylor expanded in x around 0 100.0%

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

   1. +-commutative 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

8. Final simplification 100.0%

   \[\leadsto \frac{\frac{1}{x} + 3}{\frac{1}{x} - x} \]

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ 1.0 x)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 0.0) (- (/ -3.0 x) (/ (/ 1.0 x) x)) t_0)))

C

double code(double x) {
	double t_0 = (x / (1.0 + x)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-3.0 / x) - ((1.0 / x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (1.0d0 + x)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 0.0d0) then
        tmp = ((-3.0d0) / x) - ((1.0d0 / x) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (1.0 + x)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-3.0 / x) - ((1.0 / x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (1.0 + x)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-3.0 / x) - ((1.0 / x) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(1.0 + x)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-3.0 / x) - Float64(Float64(1.0 / x) / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (1.0 + x)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-3.0 / x) - ((1.0 / x) / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(-3.0 / x), $MachinePrecision] - N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 0.0

   1. Initial program 6.7%

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

   2. Taylor expanded in x around inf 99.4%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. distribute-neg-in 99.4%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      2. unsub-neg 99.4%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      3. associate-*r/ 99.7%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      4. metadata-eval 99.7%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      5. distribute-neg-frac 99.7%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.7%

         \[\leadsto \frac{-3}{x} - \color{blue}{\sqrt{\frac{1}{{x}^{2}}} \cdot \sqrt{\frac{1}{{x}^{2}}}} \]

      2. sqrt-div 99.7%

         \[\leadsto \frac{-3}{x} - \color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{-3}{x} - \frac{\color{blue}{1}}{\sqrt{{x}^{2}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      4. unpow2 99.7%

         \[\leadsto \frac{-3}{x} - \frac{1}{\sqrt{\color{blue}{x \cdot x}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      5. sqrt-prod 44.3%

         \[\leadsto \frac{-3}{x} - \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      6. add-sqr-sqrt 99.7%

         \[\leadsto \frac{-3}{x} - \frac{1}{\color{blue}{x}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      7. sqrt-div 99.7%

         \[\leadsto \frac{-3}{x} - \frac{1}{x} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{-3}{x} - \frac{1}{x} \cdot \frac{\color{blue}{1}}{\sqrt{{x}^{2}}} \]

      9. unpow2 99.7%

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

      10. sqrt-prod 44.3%

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

      11. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. un-div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 / x) - ((1.0 / x) / x);
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	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 = ((-3.0d0) / x) - ((1.0d0 / x) / x)
    else
        tmp = 1.0d0 + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 / x) - ((1.0 / x) / x);
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-3.0 / x) - ((1.0 / x) / x)
	else:
		tmp = 1.0 + (x * 3.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (-3.0 / x) - ((1.0 / x) / x);
	else
		tmp = 1.0 + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. distribute-neg-in 99.0%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      2. unsub-neg 99.0%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      3. associate-*r/ 99.3%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      4. metadata-eval 99.3%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      5. distribute-neg-frac 99.3%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

   4. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.3%

         \[\leadsto \frac{-3}{x} - \color{blue}{\sqrt{\frac{1}{{x}^{2}}} \cdot \sqrt{\frac{1}{{x}^{2}}}} \]

      2. sqrt-div 99.3%

         \[\leadsto \frac{-3}{x} - \color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.3%

         \[\leadsto \frac{-3}{x} - \frac{\color{blue}{1}}{\sqrt{{x}^{2}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      4. unpow2 99.3%

         \[\leadsto \frac{-3}{x} - \frac{1}{\sqrt{\color{blue}{x \cdot x}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      5. sqrt-prod 44.0%

         \[\leadsto \frac{-3}{x} - \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      6. add-sqr-sqrt 99.2%

         \[\leadsto \frac{-3}{x} - \frac{1}{\color{blue}{x}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      7. sqrt-div 99.2%

         \[\leadsto \frac{-3}{x} - \frac{1}{x} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{-3}{x} - \frac{1}{x} \cdot \frac{\color{blue}{1}}{\sqrt{{x}^{2}}} \]

      9. unpow2 99.2%

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

      10. sqrt-prod 44.0%

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

      11. add-sqr-sqrt 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. un-div-inv 99.3%

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

   8. Applied egg-rr 99.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

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

Alternative 4: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	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 = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = 1.0d0 + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = 1.0 + (x * 3.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = 1.0 + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. distribute-neg-in 99.0%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      2. unsub-neg 99.0%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      3. associate-*r/ 99.3%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      4. metadata-eval 99.3%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      5. distribute-neg-frac 99.3%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

   4. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.3%

         \[\leadsto \frac{-3}{x} - \color{blue}{\sqrt{\frac{1}{{x}^{2}}} \cdot \sqrt{\frac{1}{{x}^{2}}}} \]

      2. sqrt-div 99.3%

         \[\leadsto \frac{-3}{x} - \color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.3%

         \[\leadsto \frac{-3}{x} - \frac{\color{blue}{1}}{\sqrt{{x}^{2}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      4. unpow2 99.3%

         \[\leadsto \frac{-3}{x} - \frac{1}{\sqrt{\color{blue}{x \cdot x}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      5. sqrt-prod 44.0%

         \[\leadsto \frac{-3}{x} - \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      6. add-sqr-sqrt 99.2%

         \[\leadsto \frac{-3}{x} - \frac{1}{\color{blue}{x}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      7. sqrt-div 99.2%

         \[\leadsto \frac{-3}{x} - \frac{1}{x} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{-3}{x} - \frac{1}{x} \cdot \frac{\color{blue}{1}}{\sqrt{{x}^{2}}} \]

      9. unpow2 99.2%

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

      10. sqrt-prod 44.0%

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

      11. add-sqr-sqrt 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. un-div-inv 99.3%

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

      2. sub-div 99.3%

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

   8. Applied egg-rr 99.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

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

Alternative 5: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	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 = (-3.0d0) / x
    else
        tmp = 1.0d0 + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0 + (x * 3.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0 + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 98.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.4%

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

Alternative 6: 98.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0;
	}
	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 = (-3.0d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 98.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternative 7: 50.9% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023326 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))