3frac (problem 3.3.3)

Percentage Accurate: 68.8% → 99.8%

Time: 11.1s

Alternatives: 6

Speedup: 2.1×

Specification

\[\left|x\right| > 1\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

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

Fortran

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

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

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

Fortran

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

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (/ (/ 2.0 x_m) (+ x_m 1.0)) (+ x_m -1.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((2.0 / x_m) / (x_m + 1.0)) / (x_m + -1.0));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (((2.0d0 / x_m) / (x_m + 1.0d0)) / (x_m + (-1.0d0)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (((2.0 / x_m) / (x_m + 1.0)) / (x_m + -1.0));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((2.0 / x_m) / (x_m + 1.0)) / (x_m + -1.0))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64(2.0 / x_m) / Float64(x_m + 1.0)) / Float64(x_m + -1.0)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (((2.0 / x_m) / (x_m + 1.0)) / (x_m + -1.0));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(2.0 / x$95$m), $MachinePrecision] / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

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

   1. frac-sub N/A

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

   2. frac-add N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1\right), \color{blue}{\left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right)}\right) \]

4. Applied egg-rr 22.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0

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

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot {x}^{2} - \color{blue}{x \cdot 1}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(x \cdot x\right) - x \cdot 1\right)\right) \]

   3. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} - \color{blue}{x} \cdot 1\right)\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \color{blue}{x} \cdot 1\right)\right) \]

   5. rgt-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - x \cdot \left({x}^{2} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - {x}^{3} \cdot \frac{\color{blue}{1}}{{x}^{2}}\right)\right) \]

   9. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot \color{blue}{\left(1 - \frac{1}{{x}^{2}}\right)}\right)\right) \]

   10. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{1} - \frac{1}{{x}^{2}}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot {x}^{2}\right) \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)}\right)\right)\right)\right) \]

   15. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \frac{\mathsf{neg}\left(1\right)}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \frac{-1}{{\color{blue}{x}}^{2}}\right)\right)\right)\right) \]

   17. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \frac{-1}{{x}^{2}}}\right)\right)\right) \]

   18. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + {x}^{2} \cdot \frac{\mathsf{neg}\left(1\right)}{{\color{blue}{x}}^{2}}\right)\right)\right) \]

   20. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   21. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

10. Simplified 99.5%

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

    1. associate-/r* N/A

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

    2. +-commutative N/A

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

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

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

    4. associate-/r* N/A

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

    5. associate-/r* N/A

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

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{x \cdot \left(x + 1\right)}\right), \color{blue}{\left(x - 1\right)}\right) \]

    7. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{x}}{x + 1}\right), \left(\color{blue}{x} - 1\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{x}\right), \left(x + 1\right)\right), \left(\color{blue}{x} - 1\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, x\right), \left(x + 1\right)\right), \left(x - 1\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, x\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(x - 1\right)\right) \]

    11. sub-neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, x\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, x\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(x + -1\right)\right) \]

    13. +-lowering-+.f64 99.8%

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

12. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (/ 2.0 (+ -1.0 (* x_m x_m))) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((2.0 / (-1.0 + (x_m * x_m))) / x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((2.0d0 / ((-1.0d0) + (x_m * x_m))) / x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((2.0 / (-1.0 + (x_m * x_m))) / x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((2.0 / (-1.0 + (x_m * x_m))) / x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(2.0 / Float64(-1.0 + Float64(x_m * x_m))) / x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((2.0 / (-1.0 + (x_m * x_m))) / x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(2.0 / N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

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

   1. frac-sub N/A

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

   2. frac-add N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1\right), \color{blue}{\left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right)}\right) \]

4. Applied egg-rr 22.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0

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

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot {x}^{2} - \color{blue}{x \cdot 1}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(x \cdot x\right) - x \cdot 1\right)\right) \]

   3. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} - \color{blue}{x} \cdot 1\right)\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \color{blue}{x} \cdot 1\right)\right) \]

   5. rgt-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - x \cdot \left({x}^{2} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - {x}^{3} \cdot \frac{\color{blue}{1}}{{x}^{2}}\right)\right) \]

   9. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot \color{blue}{\left(1 - \frac{1}{{x}^{2}}\right)}\right)\right) \]

   10. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{1} - \frac{1}{{x}^{2}}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot {x}^{2}\right) \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)}\right)\right)\right)\right) \]

   15. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \frac{\mathsf{neg}\left(1\right)}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \frac{-1}{{\color{blue}{x}}^{2}}\right)\right)\right)\right) \]

   17. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \frac{-1}{{x}^{2}}}\right)\right)\right) \]

   18. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + {x}^{2} \cdot \frac{\mathsf{neg}\left(1\right)}{{\color{blue}{x}}^{2}}\right)\right)\right) \]

   20. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   21. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

10. Simplified 99.5%

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-/r* N/A

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

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{x \cdot x + -1}\right), \color{blue}{x}\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(x \cdot x + -1\right)\right), x\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(-1 + x \cdot x\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(-1, \left(x \cdot x\right)\right)\right), x\right) \]

    8. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

12. Applied egg-rr 99.8%

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

Alternative 3: 99.1% accurate, 1.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ 2.0 (* x_m (+ -1.0 (* x_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (2.0 / (x_m * (-1.0 + (x_m * x_m))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (2.0d0 / (x_m * ((-1.0d0) + (x_m * x_m))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (2.0 / (x_m * (-1.0 + (x_m * x_m))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (2.0 / (x_m * (-1.0 + (x_m * x_m))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(2.0 / Float64(x_m * Float64(-1.0 + Float64(x_m * x_m)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (2.0 / (x_m * (-1.0 + (x_m * x_m))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(2.0 / N[(x$95$m * N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

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

   1. frac-sub N/A

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

   2. frac-add N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1\right), \color{blue}{\left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right)}\right) \]

4. Applied egg-rr 22.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0

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

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot {x}^{2} - \color{blue}{x \cdot 1}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(x \cdot x\right) - x \cdot 1\right)\right) \]

   3. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} - \color{blue}{x} \cdot 1\right)\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \color{blue}{x} \cdot 1\right)\right) \]

   5. rgt-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - x \cdot \left({x}^{2} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot 1 - {x}^{3} \cdot \frac{\color{blue}{1}}{{x}^{2}}\right)\right) \]

   9. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot \color{blue}{\left(1 - \frac{1}{{x}^{2}}\right)}\right)\right) \]

   10. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{1} - \frac{1}{{x}^{2}}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot {x}^{2}\right) \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(1 - \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)}\right)\right)\right)\right) \]

   15. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \frac{\mathsf{neg}\left(1\right)}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + \frac{-1}{{\color{blue}{x}}^{2}}\right)\right)\right)\right) \]

   17. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \frac{-1}{{x}^{2}}}\right)\right)\right) \]

   18. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + {x}^{2} \cdot \frac{\mathsf{neg}\left(1\right)}{{\color{blue}{x}}^{2}}\right)\right)\right) \]

   20. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   21. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

10. Simplified 99.5%

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

Alternative 4: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{2}{x\_m}}{x\_m \cdot x\_m} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (/ 2.0 x_m) (* x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((2.0 / x_m) / (x_m * x_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((2.0d0 / x_m) / (x_m * x_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((2.0 / x_m) / (x_m * x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((2.0 / x_m) / (x_m * x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(2.0 / x_m) / Float64(x_m * x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((2.0 / x_m) / (x_m * x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(2.0 / x$95$m), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{3}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2\right)}{\color{blue}{{x}^{3}}} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2\right)}{x}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 99.0%

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

8. Simplified 99.0%

   \[\leadsto \frac{\color{blue}{\frac{2}{x}}}{x \cdot x} \]
9. Add Preprocessing

Alternative 5: 98.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{2}{x\_m \cdot \left(x\_m \cdot x\_m\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ 2.0 (* x_m (* x_m x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (2.0 / (x_m * (x_m * x_m)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (2.0d0 / (x_m * (x_m * x_m)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (2.0 / (x_m * (x_m * x_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (2.0 / (x_m * (x_m * x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(2.0 / Float64(x_m * Float64(x_m * x_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (2.0 / (x_m * (x_m * x_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(2.0 / N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{3}\right)}\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 98.7%

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

5. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{2}{x \cdot \left(x \cdot x\right)}} \]
6. Add Preprocessing

Alternative 6: 5.0% accurate, 5.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ -2.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (-2.0 / x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((-2.0d0) / x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (-2.0 / x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (-2.0 / x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-2.0 / x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (-2.0 / x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 4.9%

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

5. Simplified 4.9%

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

Developer Target 1: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))