Asymptote A

Percentage Accurate: 77.4% → 99.0%

Time: 4.8s

Alternatives: 5

Speedup: 0.8×

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: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-1.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * x), $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 58.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}}{{x}^{2}} - \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}}{{x}^{2}} + \color{blue}{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}}{{x}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto -1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \frac{\color{blue}{\frac{-1 \cdot \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{4}}}}{{x}^{2}} \]

      6. associate-/r* N/A

         \[\leadsto -1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \color{blue}{\frac{-1 \cdot \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{4} \cdot {x}^{2}}} \]

      7. times-frac N/A

         \[\leadsto -1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \color{blue}{\frac{-1}{{x}^{4}} \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}} \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} \cdot \left(-1 + \frac{-1}{{x}^{4}}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} \cdot \left(-1 + \frac{-1}{{x}^{4}}\right)} \]

   5. Simplified 98.9%

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

   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 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 99.4%

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

Alternative 2: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2700:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 - \frac{2}{x \cdot x}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2700.0)
   (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 (+ x -1.0)))
   (/ (- -2.0 (/ 2.0 (* x x))) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2700.0) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	} else {
		tmp = (-2.0 - (2.0 / (x * x))) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2700.0d0) then
        tmp = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / (x + (-1.0d0)))
    else
        tmp = ((-2.0d0) - (2.0d0 / (x * x))) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2700.0) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	} else {
		tmp = (-2.0 - (2.0 / (x * x))) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2700.0:
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0))
	else:
		tmp = (-2.0 - (2.0 / (x * x))) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2700.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(-2.0 - Float64(2.0 / Float64(x * x))) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2700.0)
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	else
		tmp = (-2.0 - (2.0 / (x * x))) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2700.0], N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 - N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2700

   1. Initial program 86.1%

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

   if 2700 < x

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2700:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 - \frac{2}{x \cdot x}}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 390000:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 390000.0)
   (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 (+ x -1.0)))
   (/ -2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 390000.0) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	} else {
		tmp = -2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 390000.0d0) then
        tmp = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / (x + (-1.0d0)))
    else
        tmp = (-2.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 390000.0) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	} else {
		tmp = -2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 390000.0:
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0))
	else:
		tmp = -2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 390000.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(-2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 390000.0)
		tmp = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	else
		tmp = -2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 390000.0], N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9e5

   1. Initial program 86.1%

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

   if 3.9e5 < x

   1. Initial program 50.4%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 390000:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(2.0 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.0%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 61.7

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

   5. Simplified 61.7%

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

   if 1 < x

   1. Initial program 52.3%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 97.1

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

   5. Simplified 97.1%

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

Alternative 5: 50.9% accurate, 32.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 47.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))