3frac (problem 3.3.3)

Percentage Accurate: 70.1% → 99.8%

Time: 9.3s

Alternatives: 7

Speedup: 1.4×

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: 70.1% 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

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

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 -1.0)) (- -1.0 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) / (x_m + -1.0)) / (-1.0 - 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) / (x_m + (-1.0d0))) / ((-1.0d0) - 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) / (x_m + -1.0)) / (-1.0 - 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) / (x_m + -1.0)) / (-1.0 - x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64(-2.0 / x_m) / Float64(x_m + -1.0)) / Float64(-1.0 - 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) / (x_m + -1.0)) / (-1.0 - 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[(N[(N[(-2.0 / x$95$m), $MachinePrecision] / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

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

   1. frac-sub 19.1%

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

   2. frac-add 23.6%

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

   3. *-un-lft-identity 23.6%

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

   4. fma-define 22.4%

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

   5. *-rgt-identity 22.4%

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

   6. fmm-def 22.4%

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

6. Applied egg-rr 22.4%

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

   1. fmm-undef 22.4%

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

   2. *-commutative 22.4%

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

   3. associate-*l* 22.4%

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

8. Simplified 22.4%

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

9. Taylor expanded in x around 0 98.9%

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

    1. associate-/r* 99.9%

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

    2. div-inv 99.8%

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

    3. *-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

    1. un-div-inv 99.9%

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

    2. associate-/r* 99.9%

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

13. Applied egg-rr 99.9%

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

14. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 1.4× speedup

Math

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

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 -1.0) (- -1.0 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 * ((x_m + -1.0) * (-1.0 - 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 * ((x_m + (-1.0d0)) * ((-1.0d0) - 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 * ((x_m + -1.0) * (-1.0 - 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 * ((x_m + -1.0) * (-1.0 - 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 / Float64(x_m * Float64(Float64(x_m + -1.0) * Float64(-1.0 - 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 * ((x_m + -1.0) * (-1.0 - 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 / N[(x$95$m * N[(N[(x$95$m + -1.0), $MachinePrecision] * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

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

   1. frac-sub 19.1%

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

   2. frac-add 23.6%

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

   3. *-un-lft-identity 23.6%

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

   4. fma-define 22.4%

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

   5. *-rgt-identity 22.4%

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

   6. fmm-def 22.4%

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

6. Applied egg-rr 22.4%

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

   1. fmm-undef 22.4%

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

   2. *-commutative 22.4%

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

   3. associate-*l* 22.4%

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

8. Simplified 22.4%

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

9. Taylor expanded in x around 0 98.9%

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

10. Final simplification 98.9%

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

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

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 -1.0) (- -1.0 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) / ((x_m + -1.0) * (-1.0 - 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) / ((x_m + (-1.0d0)) * ((-1.0d0) - 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) / ((x_m + -1.0) * (-1.0 - 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) / ((x_m + -1.0) * (-1.0 - x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(-2.0 / x_m) / Float64(Float64(x_m + -1.0) * Float64(-1.0 - 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) / ((x_m + -1.0) * (-1.0 - 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[(N[(-2.0 / x$95$m), $MachinePrecision] / N[(N[(x$95$m + -1.0), $MachinePrecision] * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

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

   1. frac-sub 19.1%

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

   2. frac-add 23.6%

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

   3. *-un-lft-identity 23.6%

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

   4. fma-define 22.4%

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

   5. *-rgt-identity 22.4%

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

   6. fmm-def 22.4%

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

6. Applied egg-rr 22.4%

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

   1. fmm-undef 22.4%

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

   2. *-commutative 22.4%

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

   3. associate-*l* 22.4%

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

8. Simplified 22.4%

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

9. Taylor expanded in x around 0 98.9%

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

    1. associate-/r* 99.9%

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

    2. div-inv 99.8%

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

    3. *-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

    1. un-div-inv 99.9%

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

13. Applied egg-rr 99.9%

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

14. Final simplification 99.9%

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

Alternative 4: 69.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((1.0 / (x_m + -1.0)) + (-1.0 / 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 * ((1.0d0 / (x_m + (-1.0d0))) + ((-1.0d0) / 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 * ((1.0 / (x_m + -1.0)) + (-1.0 / x_m));
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(1.0 / Float64(x_m + -1.0)) + Float64(-1.0 / 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 * ((1.0 / (x_m + -1.0)) + (-1.0 / 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[(N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in x around inf 69.3%

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

6. Final simplification 69.3%

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

Alternative 5: 68.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((-1.0 / x_m) + (1.0 / 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 * (((-1.0d0) / x_m) + (1.0d0 / 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 * ((-1.0 / x_m) + (1.0 / x_m));
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(-1.0 / x_m) + Float64(1.0 / 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 * ((-1.0 / x_m) + (1.0 / 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[(N[(-1.0 / x$95$m), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in x around inf 69.3%

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

6. Taylor expanded in x around inf 69.0%

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

7. Final simplification 69.0%

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

Alternative 6: 5.1% 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 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in x around 0 5.1%

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

6. Final simplification 5.1%

   \[\leadsto \frac{-2}{x} \]
7. Add Preprocessing

Alternative 7: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (-1.0 / 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 * ((-1.0d0) / 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 * (-1.0 / x_m);
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-1.0 / 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 * (-1.0 / 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[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in x around inf 69.3%

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

6. Taylor expanded in x around 0 5.2%

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

7. Final simplification 5.2%

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

Developer target: 99.3% 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 2024079 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (/ 2.0 (* x (- (* x x) 1.0)))

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