Asymptote A

Percentage Accurate: 78.2% → 99.9%

Time: 6.6s

Alternatives: 6

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: 78.2% 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

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (/ 2.0 (- -1.0 x_m)) (+ -1.0 x_m)))

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return (2.0 / (-1.0 - x_m)) / (-1.0 + x_m)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(2.0 / N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.8%

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

   1. frac-sub 83.3%

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

   2. associate-/r* 83.3%

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

   3. *-un-lft-identity 83.3%

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

   4. sub-neg 83.3%

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

   5. metadata-eval 83.3%

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

   6. *-rgt-identity 83.3%

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

   7. associate--l+ 83.3%

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

   8. +-commutative 83.3%

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

   9. +-commutative 83.3%

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

   10. sub-neg 83.3%

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

   11. metadata-eval 83.3%

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

4. Applied egg-rr 83.3%

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

   1. add0 83.3%

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

   2. associate--r+ 83.3%

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

   3. associate-+r- 83.3%

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

   4. metadata-eval 83.3%

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

   5. +-commutative 83.3%

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

6. Applied egg-rr 83.3%

   \[\leadsto \frac{\color{blue}{\frac{\left(x + -2\right) - x}{x + 1} + 0}}{x + -1} \]
7. Step-by-step derivation

   1. add0 83.3%

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

   2. remove-double-neg 83.3%

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

   3. metadata-eval 83.3%

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

   4. distribute-neg-in 83.3%

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

   5. +-commutative 83.3%

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

   6. distribute-neg-frac2 83.3%

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

   7. distribute-neg-frac 83.3%

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

   8. neg-sub0 83.3%

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

   9. neg-sub0 83.3%

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

   10. associate--l+ 83.3%

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

   11. sub-neg 83.3%

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

   12. +-commutative 83.3%

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

   13. associate-+r+ 99.9%

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

   14. sub-neg 99.9%

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

   15. +-inverses 99.9%

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

   16. metadata-eval 99.9%

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

   17. metadata-eval 99.9%

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

   18. sub-neg 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 98.5% accurate, 0.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55)
   (+ (- 1.0 x_m) (/ -1.0 (+ -1.0 x_m)))
   (/ (/ -2.0 x_m) (+ -1.0 x_m))))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = (1.0d0 - x_m) + ((-1.0d0) / ((-1.0d0) + x_m))
    else
        tmp = ((-2.0d0) / x_m) / ((-1.0d0) + x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (1.0 - x_m) + (-1.0 / (-1.0 + x_m));
	} else {
		tmp = (-2.0 / x_m) / (-1.0 + x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = (1.0 - x_m) + (-1.0 / (-1.0 + x_m))
	else:
		tmp = (-2.0 / x_m) / (-1.0 + x_m)
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = (1.0 - x_m) + (-1.0 / (-1.0 + x_m));
	else
		tmp = (-2.0 / x_m) / (-1.0 + x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(N[(1.0 - x$95$m), $MachinePrecision] + N[(-1.0 / N[(-1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x$95$m), $MachinePrecision] / N[(-1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 71.6%

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

      1. neg-mul-1 71.6%

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

      2. sub-neg 71.6%

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

   5. Simplified 71.6%

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

   if 1.55000000000000004 < x

   1. Initial program 64.3%

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

      1. frac-sub 65.1%

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

      2. associate-/r* 65.1%

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

      3. *-un-lft-identity 65.1%

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

      4. sub-neg 65.1%

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

      5. metadata-eval 65.1%

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

      6. *-rgt-identity 65.1%

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

      7. associate--l+ 65.1%

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

      8. +-commutative 65.1%

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

      9. +-commutative 65.1%

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

      10. sub-neg 65.1%

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

      11. metadata-eval 65.1%

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

   4. Applied egg-rr 65.1%

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

   5. Taylor expanded in x around inf 97.7%

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

4. Final simplification 79.6%

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

Alternative 3: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x\_m}}{-1 + x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.75) 2.0 (/ (/ -2.0 x_m) (+ -1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x_m) / (-1.0 + x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.75d0) then
        tmp = 2.0d0
    else
        tmp = ((-2.0d0) / x_m) / ((-1.0d0) + x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x_m) / (-1.0 + x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.75:
		tmp = 2.0
	else:
		tmp = (-2.0 / x_m) / (-1.0 + x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0 / x_m) / Float64(-1.0 + x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.75)
		tmp = 2.0;
	else
		tmp = (-2.0 / x_m) / (-1.0 + x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.75], 2.0, N[(N[(-2.0 / x$95$m), $MachinePrecision] / N[(-1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 71.8%

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

   if 0.75 < x

   1. Initial program 64.3%

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

      1. frac-sub 65.1%

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

      2. associate-/r* 65.1%

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

      3. *-un-lft-identity 65.1%

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

      4. sub-neg 65.1%

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

      5. metadata-eval 65.1%

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

      6. *-rgt-identity 65.1%

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

      7. associate--l+ 65.1%

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

      8. +-commutative 65.1%

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

      9. +-commutative 65.1%

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

      10. sub-neg 65.1%

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

      11. metadata-eval 65.1%

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

   4. Applied egg-rr 65.1%

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

   5. Taylor expanded in x around inf 97.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{-1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 2.0 (* (+ x_m 1.0) (- 1.0 x_m))))

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0 / ((x_m + 1.0) * (1.0 - x_m))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 / N[(N[(x$95$m + 1.0), $MachinePrecision] * N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.8%

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

   1. frac-sub 83.3%

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

   2. associate-/r* 83.3%

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

   3. *-un-lft-identity 83.3%

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

   4. sub-neg 83.3%

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

   5. metadata-eval 83.3%

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

   6. *-rgt-identity 83.3%

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

   7. associate--l+ 83.3%

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

   8. +-commutative 83.3%

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

   9. +-commutative 83.3%

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

   10. sub-neg 83.3%

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

   11. metadata-eval 83.3%

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

4. Applied egg-rr 83.3%

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

   1. div-inv 83.3%

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

   2. metadata-eval 83.3%

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

   3. sub-neg 83.3%

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

   4. associate-/l* 83.3%

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

   5. associate--r+ 83.3%

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

   6. associate-+r- 83.3%

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

   7. metadata-eval 83.3%

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

   8. +-commutative 83.3%

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

   9. sub-neg 83.3%

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

   10. metadata-eval 83.3%

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

6. Applied egg-rr 83.3%

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

   1. associate-*r/ 83.3%

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

   2. associate-*r/ 83.3%

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

   3. *-rgt-identity 83.3%

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

   4. +-commutative 83.3%

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

   5. metadata-eval 83.3%

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

   6. remove-double-neg 83.3%

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

   7. distribute-neg-in 83.3%

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

   8. sub-neg 83.3%

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

   9. distribute-neg-frac2 83.3%

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

   10. associate-/l/ 83.3%

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

   11. *-commutative 83.3%

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

   12. distribute-neg-frac 83.3%

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

   13. associate--l+ 83.3%

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

   14. sub-neg 83.3%

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

   15. +-commutative 83.3%

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

   16. associate-+r+ 99.4%

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

   17. sub-neg 99.4%

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

   18. +-inverses 99.4%

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

   19. metadata-eval 99.4%

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

   20. metadata-eval 99.4%

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

8. Simplified 99.4%

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

Alternative 5: 51.3% accurate, 11.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 2.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0;
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 2.0

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in x around 0 50.8%

   \[\leadsto \color{blue}{2} \]
4. Add Preprocessing

Alternative 6: 10.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in x around 0 50.2%

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

4. Taylor expanded in x around inf 10.8%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))