3frac (problem 3.3.3)

Percentage Accurate: 68.9% → 99.8%

Time: 13.0s

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: 68.9% 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 67.7%

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

   1. +-commutative 67.7%

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

   2. associate-+r- 67.7%

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

   3. sub-neg 67.7%

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

   4. remove-double-neg 67.7%

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

   5. neg-sub0 67.7%

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

   6. associate-+l- 67.7%

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

   7. neg-sub0 67.7%

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

   8. distribute-neg-frac2 67.7%

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

   9. distribute-frac-neg2 67.7%

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

   10. associate-+r+ 67.7%

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

   11. +-commutative 67.7%

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

   12. remove-double-neg 67.7%

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

   13. distribute-neg-frac2 67.7%

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

   14. sub0-neg 67.7%

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

   15. associate-+l- 67.7%

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

   16. neg-sub0 67.7%

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

3. Simplified 67.7%

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

   1. +-commutative 67.7%

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

   2. frac-sub 22.0%

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

   3. div-inv 18.3%

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

   4. fma-define 7.3%

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

   5. *-rgt-identity 7.3%

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

   6. fma-neg 7.3%

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

6. Applied egg-rr 7.3%

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

   1. fma-undefine 18.3%

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

   2. +-commutative 18.3%

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

   3. associate-*r/ 22.0%

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

   4. *-rgt-identity 22.0%

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

   5. fma-neg 22.0%

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

8. Simplified 22.0%

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

   1. +-commutative 22.0%

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

   2. frac-add 21.4%

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

   3. *-commutative 21.4%

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

   4. associate-*r* 21.4%

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

10. Applied egg-rr 21.4%

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

11. Taylor expanded in x around 0 98.4%

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

    1. *-un-lft-identity 98.4%

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

    2. *-commutative 98.4%

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

    3. associate-/r* 99.9%

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

    4. *-commutative 99.9%

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

    5. associate-/r* 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

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 (/ (/ 1.0 (+ x_m -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 * (-2.0 * ((1.0 / (x_m + -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 * ((-2.0d0) * ((1.0d0 / (x_m + (-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 * (-2.0 * ((1.0 / (x_m + -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 * (-2.0 * ((1.0 / (x_m + -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(-2.0 * Float64(Float64(1.0 / Float64(x_m + -1.0)) / Float64(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 * (-2.0 * ((1.0 / (x_m + -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[(-2.0 * N[(N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

   1. +-commutative 67.7%

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

   2. associate-+r- 67.7%

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

   3. sub-neg 67.7%

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

   4. remove-double-neg 67.7%

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

   5. neg-sub0 67.7%

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

   6. associate-+l- 67.7%

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

   7. neg-sub0 67.7%

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

   8. distribute-neg-frac2 67.7%

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

   9. distribute-frac-neg2 67.7%

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

   10. associate-+r+ 67.7%

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

   11. +-commutative 67.7%

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

   12. remove-double-neg 67.7%

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

   13. distribute-neg-frac2 67.7%

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

   14. sub0-neg 67.7%

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

   15. associate-+l- 67.7%

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

   16. neg-sub0 67.7%

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

3. Simplified 67.7%

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

   1. +-commutative 67.7%

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

   2. frac-sub 22.0%

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

   3. div-inv 18.3%

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

   4. fma-define 7.3%

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

   5. *-rgt-identity 7.3%

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

   6. fma-neg 7.3%

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

6. Applied egg-rr 7.3%

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

   1. fma-undefine 18.3%

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

   2. +-commutative 18.3%

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

   3. associate-*r/ 22.0%

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

   4. *-rgt-identity 22.0%

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

   5. fma-neg 22.0%

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

8. Simplified 22.0%

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

   1. +-commutative 22.0%

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

   2. frac-add 21.4%

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

   3. *-commutative 21.4%

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

   4. associate-*r* 21.4%

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

10. Applied egg-rr 21.4%

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

11. Taylor expanded in x around 0 98.4%

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

    1. clear-num 98.4%

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

    2. associate-/r/ 98.4%

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

    3. associate-*l* 98.4%

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

    4. *-rgt-identity 98.4%

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

    5. associate-/r* 99.8%

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

    6. *-rgt-identity 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

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

Alternative 3: 99.2% accurate, 1.4× speedup

Math

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

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 (* (- -1.0 x_m) (* x_m (+ 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 / ((-1.0 - x_m) * (x_m * (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) / (((-1.0d0) - x_m) * (x_m * (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 / ((-1.0 - x_m) * (x_m * (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 / ((-1.0 - x_m) * (x_m * (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(-2.0 / Float64(Float64(-1.0 - x_m) * Float64(x_m * 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 / ((-1.0 - x_m) * (x_m * (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[(-2.0 / N[(N[(-1.0 - x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

   1. +-commutative 67.7%

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

   2. associate-+r- 67.7%

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

   3. sub-neg 67.7%

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

   4. remove-double-neg 67.7%

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

   5. neg-sub0 67.7%

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

   6. associate-+l- 67.7%

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

   7. neg-sub0 67.7%

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

   8. distribute-neg-frac2 67.7%

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

   9. distribute-frac-neg2 67.7%

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

   10. associate-+r+ 67.7%

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

   11. +-commutative 67.7%

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

   12. remove-double-neg 67.7%

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

   13. distribute-neg-frac2 67.7%

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

   14. sub0-neg 67.7%

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

   15. associate-+l- 67.7%

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

   16. neg-sub0 67.7%

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

3. Simplified 67.7%

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

   1. +-commutative 67.7%

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

   2. frac-sub 22.0%

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

   3. div-inv 18.3%

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

   4. fma-define 7.3%

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

   5. *-rgt-identity 7.3%

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

   6. fma-neg 7.3%

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

6. Applied egg-rr 7.3%

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

   1. fma-undefine 18.3%

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

   2. +-commutative 18.3%

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

   3. associate-*r/ 22.0%

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

   4. *-rgt-identity 22.0%

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

   5. fma-neg 22.0%

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

8. Simplified 22.0%

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

   1. +-commutative 22.0%

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

   2. frac-add 21.4%

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

   3. *-commutative 21.4%

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

   4. associate-*r* 21.4%

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

10. Applied egg-rr 21.4%

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

11. Taylor expanded in x around 0 98.4%

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

12. Final simplification 98.4%

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

Alternative 4: 67.8% accurate, 1.7× speedup

Math

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

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 67.7%

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

   1. +-commutative 67.7%

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

   2. associate-+r- 67.7%

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

   3. sub-neg 67.7%

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

   4. remove-double-neg 67.7%

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

   5. neg-sub0 67.7%

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

   6. associate-+l- 67.7%

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

   7. neg-sub0 67.7%

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

   8. distribute-neg-frac2 67.7%

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

   9. distribute-frac-neg2 67.7%

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

   10. associate-+r+ 67.7%

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

   11. +-commutative 67.7%

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

   12. remove-double-neg 67.7%

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

   13. distribute-neg-frac2 67.7%

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

   14. sub0-neg 67.7%

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

   15. associate-+l- 67.7%

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

   16. neg-sub0 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 66.3%

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

6. Final simplification 66.3%

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

Alternative 5: 67.4% 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 67.7%

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

   1. +-commutative 67.7%

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

   2. associate-+r- 67.7%

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

   3. sub-neg 67.7%

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

   4. remove-double-neg 67.7%

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

   5. neg-sub0 67.7%

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

   6. associate-+l- 67.7%

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

   7. neg-sub0 67.7%

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

   8. distribute-neg-frac2 67.7%

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

   9. distribute-frac-neg2 67.7%

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

   10. associate-+r+ 67.7%

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

   11. +-commutative 67.7%

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

   12. remove-double-neg 67.7%

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

   13. distribute-neg-frac2 67.7%

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

   14. sub0-neg 67.7%

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

   15. associate-+l- 67.7%

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

   16. neg-sub0 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 66.3%

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

6. Taylor expanded in x around inf 66.0%

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

7. Final simplification 66.0%

   \[\leadsto \frac{-1}{x} + \frac{1}{x} \]
8. 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 67.7%

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

   1. +-commutative 67.7%

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

   2. associate-+r- 67.7%

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

   3. sub-neg 67.7%

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

   4. remove-double-neg 67.7%

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

   5. neg-sub0 67.7%

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

   6. associate-+l- 67.7%

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

   7. neg-sub0 67.7%

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

   8. distribute-neg-frac2 67.7%

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

   9. distribute-frac-neg2 67.7%

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

   10. associate-+r+ 67.7%

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

   11. +-commutative 67.7%

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

   12. remove-double-neg 67.7%

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

   13. distribute-neg-frac2 67.7%

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

   14. sub0-neg 67.7%

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

   15. associate-+l- 67.7%

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

   16. neg-sub0 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around 0 4.9%

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

6. Final simplification 4.9%

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

Alternative 7: 5.0% accurate, 5.0× speedup

Math

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

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 67.7%

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

   1. +-commutative 67.7%

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

   2. associate-+r- 67.7%

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

   3. sub-neg 67.7%

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

   4. remove-double-neg 67.7%

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

   5. neg-sub0 67.7%

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

   6. associate-+l- 67.7%

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

   7. neg-sub0 67.7%

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

   8. distribute-neg-frac2 67.7%

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

   9. distribute-frac-neg2 67.7%

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

   10. associate-+r+ 67.7%

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

   11. +-commutative 67.7%

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

   12. remove-double-neg 67.7%

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

   13. distribute-neg-frac2 67.7%

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

   14. sub0-neg 67.7%

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

   15. associate-+l- 67.7%

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

   16. neg-sub0 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 66.3%

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

6. Taylor expanded in x around 0 4.9%

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

7. Final simplification 4.9%

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

Developer target: 99.2% 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 2024071 
(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))))