Asymptote A

Percentage Accurate: 77.4% → 99.9%

Time: 3.8s

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: 77.4% 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 79.0%

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

   1. frac-sub 79.4%

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

   2. associate-/r* 79.4%

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

   3. *-un-lft-identity 79.4%

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

   4. *-rgt-identity 79.4%

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

   5. associate--l- 79.4%

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

   6. +-commutative 79.4%

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

   7. +-commutative 79.4%

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

   8. sub-neg 79.4%

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

   9. metadata-eval 79.4%

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

3. Applied egg-rr 79.4%

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

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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (+ (- 1.0 x) (/ -1.0 (+ x -1.0))) (/ (/ (- 2.0) x) x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = (1.0 - x) + (-1.0 / (x + -1.0));
	} 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 <= 1.55d0) then
        tmp = (1.0d0 - x) + ((-1.0d0) / (x + (-1.0d0)))
    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 <= 1.55) {
		tmp = (1.0 - x) + (-1.0 / (x + -1.0));
	} else {
		tmp = (-2.0 / x) / x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = (1.0 - x) + (-1.0 / (x + -1.0))
	else:
		tmp = (-2.0 / x) / x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(Float64(-2.0) / x) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = (1.0 - x) + (-1.0 / (x + -1.0));
	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, 1.55], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-2.0) / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 84.4%

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

   2. Taylor expanded in x around 0 68.8%

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

      1. neg-mul-1 68.8%

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

      2. unsub-neg 68.8%

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

   4. Simplified 68.8%

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

   if 1.55000000000000004 < x

   1. Initial program 56.1%

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

   2. Taylor expanded in x around inf 96.9%

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

      1. unpow2 96.9%

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

   4. Simplified 96.9%

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

      1. associate-/r* 98.4%

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

      2. div-inv 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. un-div-inv 98.4%

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

      2. frac-2neg 98.4%

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

      3. frac-2neg 98.4%

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

      4. metadata-eval 98.4%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 51.5%

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

      7. sqr-neg 51.5%

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

      8. sqrt-unprod 51.4%

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

      9. add-sqr-sqrt 51.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 96.8%

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

      12. sqr-neg 96.8%

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

      13. sqrt-unprod 98.1%

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

      14. add-sqr-sqrt 98.4%

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

   8. Applied egg-rr 98.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.4× 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(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 84.3%

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

   2. Taylor expanded in x around 0 69.3%

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

   if 1 < x

   1. Initial program 56.9%

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

   2. Taylor expanded in x around inf 95.3%

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

      1. unpow2 95.3%

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

   4. Simplified 95.3%

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

      1. associate-/r* 96.8%

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

      2. div-inv 96.7%

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

   6. Applied egg-rr 96.7%

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

      1. un-div-inv 96.8%

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

      2. frac-2neg 96.8%

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

      3. frac-2neg 96.8%

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

      4. metadata-eval 96.8%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 50.5%

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

      7. sqr-neg 50.5%

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

      8. sqrt-unprod 50.4%

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

      9. add-sqr-sqrt 50.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 95.3%

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

      12. sqr-neg 95.3%

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

      13. sqrt-unprod 96.5%

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

      14. add-sqr-sqrt 96.8%

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

   8. Applied egg-rr 96.8%

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

4. Final simplification 74.7%

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

Alternative 4: 98.4% 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 84.3%

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

   2. Taylor expanded in x around 0 69.3%

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

   if 1 < x

   1. Initial program 56.9%

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

   2. Taylor expanded in x around inf 95.3%

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

      1. unpow2 95.3%

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

   4. Simplified 95.3%

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

4. Final simplification 74.4%

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

Alternative 5: 50.7% 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 79.0%

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

2. Taylor expanded in x around 0 56.3%

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

3. Final simplification 56.3%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))