Asymptote A

Percentage Accurate: 77.4% → 99.9%

Time: 9.2s

Alternatives: 9

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Fortran

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

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Fortran

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

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

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 77.5%

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

   1. clear-num N/A

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

   2. frac-sub N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/r* N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\frac{x + 1}{1}}\right), \color{blue}{\left(x - 1\right)}\right) \]

4. Applied egg-rr 81.6%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{-2}, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(x, -1\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.9%

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

Alternative 2: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(2.0 + Float64(Float64(x_m * x_m) * 2.0));
	else
		tmp = Float64(Float64(-2.0 / x_m) / 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 + ((x_m * x_m) * 2.0);
	else
		tmp = (-2.0 / x_m) / 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], N[(2.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \color{blue}{\left(2 \cdot {x}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 69.1%

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

   if 1 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.1%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{x}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, x\right), x\right) \]

   7. Applied egg-rr 99.3%

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

4. Final simplification 78.0%

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

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(x_m * x_m) + 2.0);
	else
		tmp = Float64(Float64(-2.0 / x_m) / 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 = (x_m * x_m) + 2.0;
	else
		tmp = (-2.0 / x_m) / 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], N[(N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision], N[(N[(-2.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 - x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

      3. --lowering--.f64 68.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

   5. Simplified 68.5%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   8. Simplified 68.4%

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

   if 1 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.1%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{x}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, x\right), x\right) \]

   7. Applied egg-rr 99.3%

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

4. Final simplification 77.4%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(x_m * x_m) + 2.0);
	else
		tmp = Float64(-2.0 / Float64(x_m * 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 = (x_m * x_m) + 2.0;
	else
		tmp = -2.0 / (x_m * 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], N[(N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision], N[(-2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 - x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

      3. --lowering--.f64 68.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

   5. Simplified 68.5%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   8. Simplified 68.4%

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

   if 1 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.1%

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

4. Final simplification 77.1%

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

Alternative 5: 53.4% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0) (+ (* 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 = (x_m * x_m) + 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 = (x_m * x_m) + 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 = (x_m * x_m) + 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 = (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 = Float64(Float64(x_m * x_m) + 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 = (x_m * x_m) + 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], N[(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 87.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 - x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

      3. --lowering--.f64 68.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

   5. Simplified 68.5%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   8. Simplified 68.4%

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

   if 1 < x

   1. Initial program 53.1%

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

      1. clear-num N/A

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

      2. frac-sub N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\frac{x + 1}{1}}\right), \color{blue}{\left(x - 1\right)}\right) \]

   4. Applied egg-rr 61.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-2}, \mathsf{+.f64}\left(x, -1\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.9%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 6.9%

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{x}\right) \]

   10. Simplified 6.9%

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

4. Final simplification 50.4%

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

Alternative 6: 53.4% accurate, 1.4× 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 87.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 1 < x

   1. Initial program 53.1%

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

      1. clear-num N/A

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

      2. frac-sub N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\frac{x + 1}{1}}\right), \color{blue}{\left(x - 1\right)}\right) \]

   4. Applied egg-rr 61.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-2}, \mathsf{+.f64}\left(x, -1\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.9%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 6.9%

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{x}\right) \]

   10. Simplified 6.9%

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

Alternative 7: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. clear-num N/A

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

   6. frac-add N/A

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

   7. *-commutative N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

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

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x + 1}{1} + \left(x - 1\right) \cdot 1\right), \color{blue}{\left(x \cdot x - 1 \cdot 1\right)}\right) \]

4. Applied egg-rr 78.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{-2}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 8: 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 77.5%

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

3. Taylor expanded in x around 0

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

5. Simplified 49.3%

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

Alternative 9: 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 77.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(1 - x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

   3. --lowering--.f64 49.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(x, 1\right)\right)\right) \]

5. Simplified 49.3%

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

6. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. lft-mult-inverse N/A

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

   5. mul-1-neg N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

   6. unsub-neg N/A

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

   7. --lowering--.f64 10.5%

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

8. Simplified 10.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
10. Step-by-step derivation

11. Simplified 10.6%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))