Asymptote A

Percentage Accurate: 77.5% → 99.4%

Time: 8.2s

Alternatives: 7

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(1 - x\right) + x \cdot \left(1 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (- 1.0 x) (* x (- 1.0 x)))))

C

double code(double x) {
	return 2.0 / ((1.0 - x) + (x * (1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((1.0d0 - x) + (x * (1.0d0 - x)))
end function

Java

public static double code(double x) {
	return 2.0 / ((1.0 - x) + (x * (1.0 - x)));
}

Python

def code(x):
	return 2.0 / ((1.0 - x) + (x * (1.0 - x)))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(1.0 - x) + Float64(x * Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((1.0 - x) + (x * (1.0 - x)));
end

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 - x), $MachinePrecision] + N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. frac-sub 78.4%

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

   2. associate-/r* 78.4%

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

   3. *-un-lft-identity 78.4%

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

   4. *-rgt-identity 78.4%

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

   5. associate--l- 78.4%

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

   6. +-commutative 78.4%

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

   7. +-commutative 78.4%

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

   8. sub-neg 78.4%

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

   9. metadata-eval 78.4%

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

4. Applied egg-rr 78.4%

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

   1. frac-2neg 78.4%

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

   2. metadata-eval 78.4%

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

   3. sub-neg 78.4%

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

   4. div-inv 78.4%

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

   5. distribute-neg-frac 78.4%

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

   6. associate-+r+ 78.4%

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

   7. metadata-eval 78.4%

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

   8. +-commutative 78.4%

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

   9. +-commutative 78.4%

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

   10. sub-neg 78.4%

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

   11. metadata-eval 78.4%

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

   12. +-commutative 78.4%

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

   13. distribute-neg-in 78.4%

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

   14. metadata-eval 78.4%

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

   15. sub-neg 78.4%

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

   16. reciprocal-define 75.9%

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

6. Applied egg-rr 75.9%

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

   1. reciprocal-undefine 78.4%

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

   2. associate-*r/ 78.4%

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

   3. neg-sub0 78.4%

      \[\leadsto \frac{\frac{\color{blue}{0 - \left(x - \left(x + 2\right)\right)}}{x + 1} \cdot 1}{1 - x} \]

   4. associate--r+ 99.9%

      \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(x - x\right) - 2\right)}}{x + 1} \cdot 1}{1 - x} \]

   5. +-inverses 99.9%

      \[\leadsto \frac{\frac{0 - \left(\color{blue}{0} - 2\right)}{x + 1} \cdot 1}{1 - x} \]

   6. associate--r- 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(0 - 0\right) + 2}}{x + 1} \cdot 1}{1 - x} \]

   7. metadata-eval 99.9%

      \[\leadsto \frac{\frac{\color{blue}{0} + 2}{x + 1} \cdot 1}{1 - x} \]

   8. metadata-eval 99.9%

      \[\leadsto \frac{\frac{\color{blue}{2}}{x + 1} \cdot 1}{1 - x} \]

   9. *-rgt-identity 99.9%

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

   10. associate-/l/ 99.6%

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

   11. *-commutative 99.6%

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

8. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-lft-in 99.7%

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

   4. *-commutative 99.7%

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

   5. *-un-lft-identity 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{2}{\left(1 - x\right) + x \cdot \left(1 - x\right)} \]
12. Add Preprocessing

Alternative 2: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5) (- (- 1.0 x) (/ 1.0 (+ x -1.0))) (/ (/ -2.0 x) (+ x -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	} else {
		tmp = (-2.0 / x) / (x + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = (1.0d0 - x) - (1.0d0 / (x + (-1.0d0)))
    else
        tmp = ((-2.0d0) / x) / (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	} else {
		tmp = (-2.0 / x) / (x + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = (1.0 - x) - (1.0 / (x + -1.0))
	else:
		tmp = (-2.0 / x) / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(Float64(1.0 - x) - Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(-2.0 / x) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	else
		tmp = (-2.0 / x) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(N[(1.0 - x), $MachinePrecision] - N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0 61.8%

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

      1. neg-mul-1 61.8%

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

      2. sub-neg 61.8%

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

   5. Simplified 61.8%

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

   if 1.5 < x

   1. Initial program 62.9%

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

      1. frac-sub 63.8%

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

      2. associate-/r* 63.8%

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

      3. *-un-lft-identity 63.8%

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

      4. *-rgt-identity 63.8%

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

      5. associate--l- 63.8%

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

      6. +-commutative 63.8%

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

      7. +-commutative 63.8%

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

      8. sub-neg 63.8%

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

      9. metadata-eval 63.8%

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

   4. Applied egg-rr 63.8%

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

   5. Taylor expanded in x around inf 98.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.76) 2.0 (/ (/ -2.0 x) (+ x -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x) / (x + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.76d0) then
        tmp = 2.0d0
    else
        tmp = ((-2.0d0) / x) / (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x) / (x + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.76:
		tmp = 2.0
	else:
		tmp = (-2.0 / x) / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0 / x) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = (-2.0 / x) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.76], 2.0, N[(N[(-2.0 / x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.76000000000000001

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{2} \]

   if 0.76000000000000001 < x

   1. Initial program 62.9%

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

      1. frac-sub 63.8%

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

      2. associate-/r* 63.8%

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

      3. *-un-lft-identity 63.8%

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

      4. *-rgt-identity 63.8%

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

      5. associate--l- 63.8%

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

      6. +-commutative 63.8%

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

      7. +-commutative 63.8%

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

      8. sub-neg 63.8%

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

      9. metadata-eval 63.8%

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

   4. Applied egg-rr 63.8%

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

   5. Taylor expanded in x around inf 98.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ (- 2.0) (+ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (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 = 2.0d0
    else
        tmp = -2.0d0 / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (1.0 + x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = -2.0 / (1.0 + x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = -2.0 / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], 2.0, N[((-2.0) / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 62.9%

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

      1. frac-sub 63.8%

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

      2. *-un-lft-identity 63.8%

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

      3. *-rgt-identity 63.8%

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

      4. associate--l- 63.8%

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

      5. +-commutative 63.8%

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

      6. difference-of-sqr-1 63.8%

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

      7. fma-neg 63.8%

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

      8. metadata-eval 63.8%

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

   4. Applied egg-rr 63.8%

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

   6. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{-\left(\mathsf{reciprocal}\left(\left(x + 1\right)\right) + \mathsf{reciprocal}\left(\left(x + 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. count-2 6.8%

         \[\leadsto -\color{blue}{2 \cdot \mathsf{reciprocal}\left(\left(x + 1\right)\right)} \]

      2. reciprocal-undefine 6.8%

         \[\leadsto -2 \cdot \color{blue}{\frac{1}{x + 1}} \]

      3. associate-*r/ 6.8%

         \[\leadsto -\color{blue}{\frac{2 \cdot 1}{x + 1}} \]

      4. metadata-eval 6.8%

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

   8. Simplified 6.8%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ (- 2.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = -2.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], 2.0, N[((-2.0) / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 62.9%

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

      1. frac-sub 63.8%

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

      2. *-un-lft-identity 63.8%

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

      3. *-rgt-identity 63.8%

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

      4. associate--l- 63.8%

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

      5. +-commutative 63.8%

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

      6. difference-of-sqr-1 63.8%

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

      7. fma-neg 63.8%

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

      8. metadata-eval 63.8%

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

   4. Applied egg-rr 63.8%

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

   6. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{-\left(\mathsf{reciprocal}\left(\left(x + 1\right)\right) + \mathsf{reciprocal}\left(\left(x + 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. count-2 6.8%

         \[\leadsto -\color{blue}{2 \cdot \mathsf{reciprocal}\left(\left(x + 1\right)\right)} \]

      2. reciprocal-undefine 6.8%

         \[\leadsto -2 \cdot \color{blue}{\frac{1}{x + 1}} \]

      3. associate-*r/ 6.8%

         \[\leadsto -\color{blue}{\frac{2 \cdot 1}{x + 1}} \]

      4. metadata-eval 6.8%

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

   8. Simplified 6.8%

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

   9. Taylor expanded in x around inf 6.8%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(1 - x\right) \cdot \left(1 + x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* (- 1.0 x) (+ 1.0 x))))

C

double code(double x) {
	return 2.0 / ((1.0 - x) * (1.0 + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((1.0d0 - x) * (1.0d0 + x))
end function

Java

public static double code(double x) {
	return 2.0 / ((1.0 - x) * (1.0 + x));
}

Python

def code(x):
	return 2.0 / ((1.0 - x) * (1.0 + x))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(1.0 - x) * Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((1.0 - x) * (1.0 + x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. frac-sub 78.4%

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

   2. associate-/r* 78.4%

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

   3. *-un-lft-identity 78.4%

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

   4. *-rgt-identity 78.4%

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

   5. associate--l- 78.4%

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

   6. +-commutative 78.4%

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

   7. +-commutative 78.4%

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

   8. sub-neg 78.4%

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

   9. metadata-eval 78.4%

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

4. Applied egg-rr 78.4%

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

   1. frac-2neg 78.4%

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

   2. metadata-eval 78.4%

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

   3. sub-neg 78.4%

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

   4. div-inv 78.4%

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

   5. distribute-neg-frac 78.4%

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

   6. associate-+r+ 78.4%

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

   7. metadata-eval 78.4%

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

   8. +-commutative 78.4%

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

   9. +-commutative 78.4%

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

   10. sub-neg 78.4%

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

   11. metadata-eval 78.4%

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

   12. +-commutative 78.4%

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

   13. distribute-neg-in 78.4%

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

   14. metadata-eval 78.4%

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

   15. sub-neg 78.4%

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

   16. reciprocal-define 75.9%

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

6. Applied egg-rr 75.9%

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

   1. reciprocal-undefine 78.4%

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

   2. associate-*r/ 78.4%

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

   3. neg-sub0 78.4%

      \[\leadsto \frac{\frac{\color{blue}{0 - \left(x - \left(x + 2\right)\right)}}{x + 1} \cdot 1}{1 - x} \]

   4. associate--r+ 99.9%

      \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(x - x\right) - 2\right)}}{x + 1} \cdot 1}{1 - x} \]

   5. +-inverses 99.9%

      \[\leadsto \frac{\frac{0 - \left(\color{blue}{0} - 2\right)}{x + 1} \cdot 1}{1 - x} \]

   6. associate--r- 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(0 - 0\right) + 2}}{x + 1} \cdot 1}{1 - x} \]

   7. metadata-eval 99.9%

      \[\leadsto \frac{\frac{\color{blue}{0} + 2}{x + 1} \cdot 1}{1 - x} \]

   8. metadata-eval 99.9%

      \[\leadsto \frac{\frac{\color{blue}{2}}{x + 1} \cdot 1}{1 - x} \]

   9. *-rgt-identity 99.9%

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

   10. associate-/l/ 99.6%

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

   11. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \frac{2}{\left(1 - x\right) \cdot \left(1 + x\right)} \]
10. Add Preprocessing

Alternative 7: 50.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in x around 0 46.3%

   \[\leadsto \color{blue}{2} \]

4. Final simplification 46.3%

   \[\leadsto 2 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))