Asymptote A

Percentage Accurate: 76.5% → 99.9%

Time: 6.7s

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: 76.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-sub N/A

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

   9. associate-/r* N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. frac-2neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 84.4%

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

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%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[(1.0 / N[(1.0 + x$95$m), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 2.0 + 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 58.2%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 95.9

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

   5. Applied rewrites 95.9%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 97.7%

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

Alternative 3: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. *-lft-identity N/A

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

   11. *-rgt-identity N/A

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

   12. lift-+.f64 N/A

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

   13. associate--r+ N/A

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

   14. lower--.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 84.4

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

4. Applied rewrites 84.4%

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

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 99.3%

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

   1. lift-fma.f64 N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. neg-mul-1 N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. lift--.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. lift-fma.f64 N/A

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

    2. lift--.f64 N/A

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

    3. associate-+r- N/A

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

    4. lower--.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 4: 52.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma((x_m * x_m), 2.0, 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(Float64(x_m * x_m), 2.0, 2.0);
	else
		tmp = Float64(2.0 / Float64(-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[(N[(x$95$m * x$95$m), $MachinePrecision] * 2.0 + 2.0), $MachinePrecision], N[(2.0 / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 89.5%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 1 < x

   1. Initial program 54.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. associate-/r* N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. frac-2neg N/A

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

      17. div-inv N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 6.9%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.9

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

   10. Applied rewrites 6.9%

       \[\leadsto \frac{2}{\color{blue}{-x}} \]
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 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. *-lft-identity N/A

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

   11. *-rgt-identity N/A

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

   12. lift-+.f64 N/A

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

   13. associate--r+ N/A

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

   14. lower--.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 84.4

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

4. Applied rewrites 84.4%

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

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 99.3%

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

Alternative 6: 51.7% 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 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-sub N/A

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

   9. associate-/r* N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. frac-2neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 84.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 55.5%

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

Alternative 7: 50.0% 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 80.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.7%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))