Asymptote A

Percentage Accurate: 77.7% → 99.9%

Time: 4.0s

Alternatives: 5

Speedup: 1.6×

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: 77.7% 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: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{-2}{1 + x}}{x + -1} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ (/ -2.0 (+ 1.0 x)) (+ x -1.0)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / (1.0d0 + x)) / (x + (-1.0d0))
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return (-2.0 / (1.0 + x)) / (x + -1.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[(-2.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. frac-sub 77.8%

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

   2. associate-/r* 77.8%

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

   3. *-un-lft-identity 77.8%

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

   4. *-rgt-identity 77.8%

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

   5. associate--l- 77.8%

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

   6. +-commutative 77.8%

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

   7. +-commutative 77.8%

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

   8. sub-neg 77.8%

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

   9. metadata-eval 77.8%

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

3. Applied egg-rr 77.8%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.3% accurate, 1.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ -2.0 (* x x))))

C

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

Fortran

NOTE: x should be positive before calling this function
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 * x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.0], 2.0, N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 81.3%

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

   2. Taylor expanded in x around 0 65.9%

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

   if 1 < x

   1. Initial program 66.6%

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

   2. Taylor expanded in x around inf 98.2%

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

      1. unpow2 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 74.9%

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

Alternative 3: 98.8% accurate, 1.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ (/ -2.0 x) x)))

C

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

Fortran

NOTE: x should be positive before calling this function
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) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.0], 2.0, N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 81.3%

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

   2. Taylor expanded in x around 0 65.9%

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

   if 1 < x

   1. Initial program 66.6%

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

      1. frac-sub 66.8%

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

      2. associate-/r* 66.8%

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

      3. *-un-lft-identity 66.8%

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

      4. *-rgt-identity 66.8%

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

      5. associate--l- 66.8%

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

      6. +-commutative 66.8%

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

      7. +-commutative 66.8%

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

      8. sub-neg 66.8%

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

      9. metadata-eval 66.8%

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

   3. Applied egg-rr 66.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. unpow2 98.2%

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

      2. associate-/r* 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 74.8%

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

Alternative 4: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{2}{1 - x \cdot x} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ 2.0 (- 1.0 (* x x))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (1.0d0 - (x * x))
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 2.0 / (1.0 - (x * x));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(2.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. frac-2neg 77.2%

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

   2. metadata-eval 77.2%

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

   3. frac-sub 77.8%

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

   4. *-un-lft-identity 77.8%

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

   5. sub-neg 77.8%

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

   6. metadata-eval 77.8%

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

   7. distribute-neg-in 77.8%

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

   8. metadata-eval 77.8%

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

   9. +-commutative 77.8%

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

   10. +-commutative 77.8%

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

   11. sub-neg 77.8%

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

   12. metadata-eval 77.8%

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

   13. distribute-neg-in 77.8%

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

   14. metadata-eval 77.8%

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

3. Applied egg-rr 77.8%

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

4. Taylor expanded in x around 0 99.8%

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

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \frac{2}{\color{blue}{1 + -1 \cdot {x}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 99.8%

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

   2. mul-1-neg 99.8%

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

   3. unsub-neg 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 50.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 2 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 2.0)

C

x = abs(x);
double code(double x) {
	return 2.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 2.0;
}

Python

x = abs(x)
def code(x):
	return 2.0

Julia

x = abs(x)
function code(x)
	return 2.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 2.0

Derivation

1. Initial program 77.2%

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

2. Taylor expanded in x around 0 48.4%

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

3. Final simplification 48.4%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023213 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))