Asymptote A

Percentage Accurate: 77.4% → 99.9%

Time: 7.9s

Alternatives: 7

Speedup: 1.8×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (-2.0 / (-1.0 + x_m)) / (x_m + 1.0);
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[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. lift-+.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. frac-sub N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 76.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

Alternative 2: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1}{x\_m + 1} + \frac{-1}{-1 + x\_m} \leq 0:\\ \;\;\;\;\frac{-2}{x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x\_m \cdot x\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (/ 1.0 (+ x_m 1.0)) (/ -1.0 (+ -1.0 x_m))) 0.0)
   (/ -2.0 (* x_m x_m))
   (fma 2.0 (* x_m x_m) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (((1.0 / (x_m + 1.0)) + (-1.0 / (-1.0 + x_m))) <= 0.0) {
		tmp = -2.0 / (x_m * x_m);
	} else {
		tmp = fma(2.0, (x_m * x_m), 2.0);
	}
	return tmp;
}

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(-1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 0.0

   1. Initial program 51.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot {x}^{2} + 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{2}, 2\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{x \cdot x}, 2\right) \]

      4. lower-*.f64 99.6

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{x \cdot x}, 2\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x \cdot x, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{-1 + x} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x \cdot x, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma(2.0, (x_m * x_m), 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 = fma(2.0, Float64(x_m * x_m), 2.0);
	else
		tmp = Float64(-2.0 / x_m);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot {x}^{2} + 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{2}, 2\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{x \cdot x}, 2\right) \]

      4. lower-*.f64 70.4

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{x \cdot x}, 2\right) \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x \cdot x, 2\right)} \]

   if 1 < x

   1. Initial program 53.6%

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

      1. lift-+.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. frac-sub N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 6.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 6.6

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

   10. Applied rewrites 6.6%

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

Alternative 4: 53.2% accurate, 1.8× 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 84.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.4%

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

   if 1 < x

   1. Initial program 53.6%

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

      1. lift-+.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. frac-sub N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 6.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 6.6

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

   10. Applied rewrites 6.6%

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

Alternative 5: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-+.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. frac-add N/A

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

   16. *-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. lift--.f64 N/A

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

   19. difference-of-sqr-1 N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 N/A

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

4. Applied rewrites 76.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
8. Add Preprocessing

Alternative 6: 52.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-+.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. frac-sub N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 76.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 53.5%

   \[\leadsto \frac{\color{blue}{2}}{1 - x} \]
8. Add Preprocessing

Alternative 7: 51.1% accurate, 32.0× 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 76.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2} \]
4. Step-by-step derivation

5. Applied rewrites 52.4%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))