Asymptote A

Percentage Accurate: 76.9% → 99.9%

Time: 7.0s

Alternatives: 8

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: 76.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. frac-sub 82.8%

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

   2. *-commutative 82.8%

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

   3. associate-/r* 82.8%

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

   4. *-rgt-identity 82.8%

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

   5. *-un-lft-identity 82.8%

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

   6. sub-neg 82.8%

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

   7. associate--l+ 82.8%

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

   8. +-commutative 82.8%

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

   9. associate--r+ 82.8%

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

   10. metadata-eval 82.8%

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

   11. metadata-eval 82.8%

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

   12. sub-neg 82.8%

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

   13. metadata-eval 82.8%

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

   14. +-commutative 82.8%

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

3. Applied egg-rr 82.8%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.8:\\ \;\;\;\;\left(1 - x_m\right) + \frac{-1}{x_m + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x_m}}{x_m + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.8)
   (+ (- 1.0 x_m) (/ -1.0 (+ x_m -1.0)))
   (/ (/ -2.0 x_m) (+ x_m 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.8) {
		tmp = (1.0 - x_m) + (-1.0 / (x_m + -1.0));
	} else {
		tmp = (-2.0 / x_m) / (x_m + 1.0);
	}
	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 <= 1.8d0) then
        tmp = (1.0d0 - x_m) + ((-1.0d0) / (x_m + (-1.0d0)))
    else
        tmp = ((-2.0d0) / x_m) / (x_m + 1.0d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.8) {
		tmp = (1.0 - x_m) + (-1.0 / (x_m + -1.0));
	} else {
		tmp = (-2.0 / x_m) / (x_m + 1.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.8:
		tmp = (1.0 - x_m) + (-1.0 / (x_m + -1.0))
	else:
		tmp = (-2.0 / x_m) / (x_m + 1.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.8)
		tmp = Float64(Float64(1.0 - x_m) + Float64(-1.0 / Float64(x_m + -1.0)));
	else
		tmp = Float64(Float64(-2.0 / x_m) / Float64(x_m + 1.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.8)
		tmp = (1.0 - x_m) + (-1.0 / (x_m + -1.0));
	else
		tmp = (-2.0 / x_m) / (x_m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.8], N[(N[(1.0 - x$95$m), $MachinePrecision] + N[(-1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x$95$m), $MachinePrecision] / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 90.9%

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

   2. Taylor expanded in x around 0 75.2%

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

      1. mul-1-neg 75.2%

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

      2. sub-neg 75.2%

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

   4. Simplified 75.2%

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

   if 1.80000000000000004 < x

   1. Initial program 53.6%

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

      1. frac-sub 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-/r* 56.6%

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

      4. *-rgt-identity 56.6%

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

      5. *-un-lft-identity 56.6%

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

      6. sub-neg 56.6%

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

      7. associate--l+ 56.6%

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

      8. +-commutative 56.6%

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

      9. associate--r+ 56.6%

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

      10. metadata-eval 56.6%

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

      11. metadata-eval 56.6%

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

      12. sub-neg 56.6%

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

      13. metadata-eval 56.6%

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

      14. +-commutative 56.6%

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

   3. Applied egg-rr 56.6%

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

   4. Taylor expanded in x around inf 97.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + 1}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 90.9%

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

   2. Taylor expanded in x around 0 75.3%

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

   if 1 < x

   1. Initial program 53.6%

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

      1. frac-sub 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-/r* 56.6%

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

      4. *-rgt-identity 56.6%

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

      5. *-un-lft-identity 56.6%

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

      6. sub-neg 56.6%

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

      7. associate--l+ 56.6%

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

      8. +-commutative 56.6%

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

      9. associate--r+ 56.6%

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

      10. metadata-eval 56.6%

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

      11. metadata-eval 56.6%

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

      12. sub-neg 56.6%

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

      13. metadata-eval 56.6%

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

      14. +-commutative 56.6%

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

   3. Applied egg-rr 56.6%

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

   4. Taylor expanded in x around inf 97.8%

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

4. Final simplification 80.8%

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

Alternative 4: 99.4% 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 81.9%

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

   1. frac-sub 82.8%

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

   2. *-commutative 82.8%

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

   3. associate-/r* 82.8%

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

   4. *-rgt-identity 82.8%

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

   5. *-un-lft-identity 82.8%

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

   6. sub-neg 82.8%

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

   7. associate--l+ 82.8%

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

   8. +-commutative 82.8%

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

   9. associate--r+ 82.8%

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

   10. metadata-eval 82.8%

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

   11. metadata-eval 82.8%

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

   12. sub-neg 82.8%

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

   13. metadata-eval 82.8%

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

   14. +-commutative 82.8%

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

3. Applied egg-rr 82.8%

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

4. Taylor expanded in x around 0 99.9%

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

   1. div-inv 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. frac-2neg 99.9%

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

   6. metadata-eval 99.9%

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

   7. distribute-neg-in 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

   1. associate-/l/ 99.2%

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

   2. associate-*r/ 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 5: 53.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 90.9%

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

   2. Taylor expanded in x around 0 75.3%

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

   if 1 < x

   1. Initial program 53.6%

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

      1. frac-2neg 53.6%

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

      2. frac-sub 56.6%

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

      3. distribute-lft-neg-in 56.6%

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

      4. *-un-lft-identity 56.6%

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

      5. sub-neg 56.6%

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

      6. distribute-neg-in 56.6%

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

      7. metadata-eval 56.6%

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

      8. metadata-eval 56.6%

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

      9. +-commutative 56.6%

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

      10. sub-neg 56.6%

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

      11. *-commutative 56.6%

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

      12. *-un-lft-identity 56.6%

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

      13. neg-sub0 56.6%

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

      14. +-commutative 56.6%

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

      15. associate--r+ 56.6%

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

      16. metadata-eval 56.6%

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

      17. *-commutative 56.6%

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

      18. sub-neg 56.6%

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

      19. metadata-eval 56.6%

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

      20. neg-sub0 56.6%

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

      21. +-commutative 56.6%

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

      22. associate--r+ 56.6%

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

      23. metadata-eval 56.6%

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

   3. Applied egg-rr 56.6%

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

      1. associate-/l/ 56.6%

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

      2. div-sub 53.9%

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

      3. *-inverses 53.9%

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

      4. sub-neg 53.9%

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

      5. metadata-eval 53.9%

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

   5. Simplified 53.9%

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

   6. Taylor expanded in x around 0 6.6%

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

   7. Taylor expanded in x around inf 6.6%

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

4. Final simplification 58.7%

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

Alternative 6: 52.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. frac-2neg 81.9%

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

   2. frac-sub 82.8%

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

   3. distribute-lft-neg-in 82.8%

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

   4. *-un-lft-identity 82.8%

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

   5. sub-neg 82.8%

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

   6. distribute-neg-in 82.8%

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

   7. metadata-eval 82.8%

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

   8. metadata-eval 82.8%

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

   9. +-commutative 82.8%

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

   10. sub-neg 82.8%

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

   11. *-commutative 82.8%

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

   12. *-un-lft-identity 82.8%

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

   13. neg-sub0 82.8%

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

   14. +-commutative 82.8%

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

   15. associate--r+ 82.8%

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

   16. metadata-eval 82.8%

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

   17. *-commutative 82.8%

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

   18. sub-neg 82.8%

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

   19. metadata-eval 82.8%

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

   20. neg-sub0 82.8%

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

   21. +-commutative 82.8%

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

   22. associate--r+ 82.8%

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

   23. metadata-eval 82.8%

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

3. Applied egg-rr 82.8%

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

   1. associate-/l/ 82.8%

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

   2. div-sub 82.0%

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

   3. *-inverses 82.0%

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

   4. sub-neg 82.0%

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

   5. metadata-eval 82.0%

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

5. Simplified 82.0%

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

6. Taylor expanded in x around 0 58.2%

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

7. Final simplification 58.2%

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

Alternative 7: 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 81.9%

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

2. Taylor expanded in x around 0 56.9%

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

3. Taylor expanded in x around inf 11.9%

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

4. Final simplification 11.9%

   \[\leadsto 1 \]

Alternative 8: 51.1% 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 81.9%

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

2. Taylor expanded in x around 0 57.7%

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

3. Final simplification 57.7%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))