Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-add 71.1%

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

   2. div-inv 71.0%

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

   3. fma-define 71.0%

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

   4. *-rgt-identity 71.0%

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

   5. +-commutative 71.0%

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

   6. +-commutative 71.0%

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

   7. *-un-lft-identity 71.0%

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

   8. fma-define 71.0%

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

   9. metadata-eval 71.0%

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

   10. fma-neg 71.0%

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

   11. *-un-lft-identity 71.0%

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

   12. difference-of-sqr-1 71.0%

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

   13. fma-neg 71.0%

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

   14. metadata-eval 71.0%

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

6. Applied egg-rr 71.0%

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

8. Simplified 71.0%

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

   1. fma-undefine 71.0%

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

   2. sqr-neg 71.0%

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

   3. mul-1-neg 71.0%

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

   4. distribute-lft-neg-out 71.0%

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

   5. add-sqr-sqrt 33.0%

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

   6. sqrt-unprod 56.9%

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

   7. mul-1-neg 56.9%

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

   8. mul-1-neg 56.9%

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

   9. sqr-neg 56.9%

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

   10. sqrt-unprod 23.8%

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

   11. add-sqr-sqrt 46.2%

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

   12. metadata-eval 46.2%

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

   13. distribute-neg-in 46.2%

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

   14. rgt-mult-inverse 46.3%

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

   15. distribute-lft-in 46.3%

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

   16. distribute-lft-neg-in 46.3%

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

   17. mul-1-neg 46.3%

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

   18. add-sqr-sqrt 22.2%

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

   19. sqrt-unprod 23.8%

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

   20. mul-1-neg 23.8%

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

   21. mul-1-neg 23.8%

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

   22. sqr-neg 23.8%

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

   23. sqrt-unprod 13.5%

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

10. Applied egg-rr 71.1%

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

    1. add-sqr-sqrt 71.0%

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

    2. times-frac 71.9%

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

    3. +-commutative 71.9%

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

    4. distribute-lft-in 71.9%

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

    5. rgt-mult-inverse 71.9%

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

    6. hypot-1-def 71.9%

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

    7. +-commutative 71.9%

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

    8. distribute-lft-in 71.9%

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

    9. rgt-mult-inverse 71.9%

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

    10. hypot-1-def 99.9%

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

12. Applied egg-rr 99.9%

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

14. Simplified 100.0%

    \[\leadsto \color{blue}{\frac{x - \frac{-1}{x}}{x + \frac{-1}{x}}} \]
15. Add Preprocessing

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   1.0
   (if (<= x 1.85) (+ x (/ 1.0 (+ x -1.0))) (+ (/ x (+ x 1.0)) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.85) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = (x / (x + 1.0)) + (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
    else if (x <= 1.85d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = (x / (x + 1.0d0)) + (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;
	} else if (x <= 1.85) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = (x / (x + 1.0)) + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.85:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = (x / (x + 1.0)) + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.85)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.85)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = (x / (x + 1.0)) + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.85], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 1.8500000000000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.8%

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

   if 1.8500000000000001 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.9) (+ x (/ 1.0 (+ x -1.0))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.9d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.9:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.9], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8999999999999999 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.1%

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

   if -1 < x < 1.8999999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 1.0) -1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.7%

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

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x}{x + 1} + \frac{1}{x + -1} \]
4. Add Preprocessing

Alternative 6: 50.2% accurate, 11.0× 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 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 46.7%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))