3frac (problem 3.3.3)

Percentage Accurate: 84.9% → 99.9%

Time: 9.8s

Alternatives: 7

Speedup: 1.4×

Specification

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: 84.9% 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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (/ -2.0 (+ x_m (pow x_m 2.0))) (- 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 + pow(x_m, 2.0))) / (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 ** 2.0d0))) / (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 + Math.pow(x_m, 2.0))) / (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 + math.pow(x_m, 2.0))) / (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 / Float64(x_m + (x_m ^ 2.0))) / 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 ^ 2.0))) / (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 / N[(x$95$m + N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. sub-neg 83.8%

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

   2. distribute-neg-frac 83.8%

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

   3. metadata-eval 83.8%

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

   4. metadata-eval 83.8%

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

   5. metadata-eval 83.8%

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

   6. associate-/r* 83.8%

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

   7. metadata-eval 83.8%

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

   8. neg-mul-1 83.8%

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

   9. +-commutative 83.8%

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

   10. associate-+l+ 83.8%

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

   11. +-commutative 83.8%

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

   12. neg-mul-1 83.8%

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

   13. metadata-eval 83.8%

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

   14. associate-/r* 83.8%

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

   15. metadata-eval 83.8%

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

   16. metadata-eval 83.8%

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

   17. +-commutative 83.8%

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

   18. +-commutative 83.8%

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

3. Simplified 83.8%

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

   1. +-commutative 83.8%

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

   2. frac-add 59.7%

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

   3. frac-add 58.8%

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

   4. *-un-lft-identity 58.8%

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

   5. *-commutative 58.8%

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

   6. neg-mul-1 58.8%

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

   7. distribute-neg-in 58.8%

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

   8. metadata-eval 58.8%

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

5. Applied egg-rr 58.8%

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

   1. expm1-log1p-u 33.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\left(1 - x\right) + \left(-1 + \left(-x\right)\right)\right) \cdot x + \left(\left(1 + x\right) \cdot \left(1 - x\right)\right) \cdot -2}{\left(\left(1 + x\right) \cdot \left(1 - x\right)\right) \cdot x}\right)\right)} \]

   2. expm1-udef 33.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(1 - x\right) + \left(-1 + \left(-x\right)\right)\right) \cdot x + \left(\left(1 + x\right) \cdot \left(1 - x\right)\right) \cdot -2}{\left(\left(1 + x\right) \cdot \left(1 - x\right)\right) \cdot x}\right)} - 1} \]

7. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 1 - \left(x - \left(-1 - x\right)\right), \left(\left(1 + x\right) \cdot \left(1 - x\right)\right) \cdot -2\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(1 - x\right)\right)}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 33.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 1 - \left(x - \left(-1 - x\right)\right), \left(\left(1 + x\right) \cdot \left(1 - x\right)\right) \cdot -2\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(1 - x\right)\right)}\right)\right)} \]

   2. expm1-log1p 58.3%

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

   3. associate-*r* 58.3%

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

   4. associate-/r* 53.2%

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

9. Simplified 58.8%

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

10. Taylor expanded in x around 0 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 84.8% accurate, 1.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (/ -2.0 x_m) (/ 2.0 (+ 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 / x_m) + (2.0 / (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) / x_m) + (2.0d0 / (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 / x_m) + (2.0 / (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 / x_m) + (2.0 / (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(-2.0 / x_m) + Float64(2.0 / 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 / x_m) + (2.0 / (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[(-2.0 / x$95$m), $MachinePrecision] + N[(2.0 / N[(x$95$m + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. sub-neg 83.8%

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

   2. distribute-neg-frac 83.8%

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

   3. metadata-eval 83.8%

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

   4. metadata-eval 83.8%

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

   5. metadata-eval 83.8%

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

   6. associate-/r* 83.8%

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

   7. metadata-eval 83.8%

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

   8. neg-mul-1 83.8%

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

   9. +-commutative 83.8%

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

   10. associate-+l+ 83.8%

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

   11. +-commutative 83.8%

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

   12. neg-mul-1 83.8%

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

   13. metadata-eval 83.8%

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

   14. associate-/r* 83.8%

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

   15. metadata-eval 83.8%

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

   16. metadata-eval 83.8%

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

   17. +-commutative 83.8%

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

   18. +-commutative 83.8%

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

3. Simplified 83.8%

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

   1. frac-add 59.7%

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

   2. clear-num 59.7%

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

   3. *-un-lft-identity 59.7%

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

   4. *-commutative 59.7%

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

   5. neg-mul-1 59.7%

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

   6. distribute-neg-in 59.7%

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

   7. metadata-eval 59.7%

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

5. Applied egg-rr 59.7%

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

6. Taylor expanded in x around 0 83.8%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{0.5 \cdot x - 0.5 \cdot \frac{1}{x}}} \]
7. Step-by-step derivation

   1. *-commutative 83.8%

      \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{x \cdot 0.5} - 0.5 \cdot \frac{1}{x}} \]

   2. associate-*r/ 83.8%

      \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \color{blue}{\frac{0.5 \cdot 1}{x}}} \]

   3. metadata-eval 83.8%

      \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{\color{blue}{0.5}}{x}} \]

8. Simplified 83.8%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{x \cdot 0.5 - \frac{0.5}{x}}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 58.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}}\right)\right)} \]

   2. expm1-udef 58.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}}\right)} - 1} \]

10. Applied egg-rr 58.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 58.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}}\right)\right)} \]

    2. expm1-log1p 83.8%

       \[\leadsto \color{blue}{\frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}}} \]

    3. metadata-eval 83.8%

       \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{\color{blue}{1 \cdot 0.5}}{x}} \]

    4. associate-*l/ 83.8%

       \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \color{blue}{\frac{1}{x} \cdot 0.5}} \]

    5. distribute-rgt-out-- 83.8%

       \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{0.5 \cdot \left(x - \frac{1}{x}\right)}} \]

    6. associate-/r* 83.8%

       \[\leadsto \frac{-2}{x} + \color{blue}{\frac{\frac{1}{0.5}}{x - \frac{1}{x}}} \]

    7. metadata-eval 83.8%

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

    8. sub-neg 83.8%

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

    9. distribute-neg-frac 83.8%

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

    10. metadata-eval 83.8%

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

12. Simplified 83.8%

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

13. Final simplification 83.8%

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

Alternative 3: 99.6% accurate, 1.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ 2.0 (* (+ x_m 1.0) (* 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 / ((x_m + 1.0) * (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 / ((x_m + 1.0d0) * (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 / ((x_m + 1.0) * (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 / ((x_m + 1.0) * (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(x_m + 1.0) * 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 / ((x_m + 1.0) * (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[(x$95$m + 1.0), $MachinePrecision] * N[(x$95$m * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. sub-neg 83.8%

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

   2. distribute-neg-frac 83.8%

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

   3. metadata-eval 83.8%

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

   4. metadata-eval 83.8%

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

   5. metadata-eval 83.8%

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

   6. associate-/r* 83.8%

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

   7. metadata-eval 83.8%

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

   8. neg-mul-1 83.8%

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

   9. +-commutative 83.8%

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

   10. associate-+l+ 83.8%

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

   11. +-commutative 83.8%

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

   12. neg-mul-1 83.8%

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

   13. metadata-eval 83.8%

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

   14. associate-/r* 83.8%

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

   15. metadata-eval 83.8%

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

   16. metadata-eval 83.8%

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

   17. +-commutative 83.8%

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

   18. +-commutative 83.8%

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

3. Simplified 83.8%

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

   1. associate-+r+ 83.8%

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

   2. +-commutative 83.8%

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

   3. metadata-eval 83.8%

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

   4. sub-neg 83.8%

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

   5. metadata-eval 83.8%

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

   6. distribute-neg-in 83.8%

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

   7. +-commutative 83.8%

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

   8. frac-2neg 83.8%

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

   9. associate-+r+ 83.8%

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

   10. frac-add 59.7%

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

   11. frac-add 58.8%

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

5. Applied egg-rr 58.8%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.9% accurate, 1.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (/ 2.0 (+ x_m 1.0)) (* 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 / (x_m + 1.0)) / (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 / (x_m + 1.0d0)) / (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 / (x_m + 1.0)) / (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 / (x_m + 1.0)) / (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(Float64(2.0 / Float64(x_m + 1.0)) / 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 / (x_m + 1.0)) / (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[(N[(2.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. sub-neg 83.8%

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

   2. distribute-neg-frac 83.8%

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

   3. metadata-eval 83.8%

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

   4. metadata-eval 83.8%

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

   5. metadata-eval 83.8%

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

   6. associate-/r* 83.8%

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

   7. metadata-eval 83.8%

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

   8. neg-mul-1 83.8%

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

   9. +-commutative 83.8%

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

   10. associate-+l+ 83.8%

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

   11. +-commutative 83.8%

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

   12. neg-mul-1 83.8%

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

   13. metadata-eval 83.8%

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

   14. associate-/r* 83.8%

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

   15. metadata-eval 83.8%

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

   16. metadata-eval 83.8%

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

   17. +-commutative 83.8%

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

   18. +-commutative 83.8%

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

3. Simplified 83.8%

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

   1. associate-+r+ 83.8%

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

   2. +-commutative 83.8%

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

   3. metadata-eval 83.8%

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

   4. sub-neg 83.8%

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

   5. metadata-eval 83.8%

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

   6. distribute-neg-in 83.8%

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

   7. +-commutative 83.8%

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

   8. frac-2neg 83.8%

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

   9. associate-+r+ 83.8%

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

   10. frac-add 59.7%

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

   11. frac-add 58.8%

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

5. Applied egg-rr 58.8%

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

6. Taylor expanded in x around 0 99.7%

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

   1. expm1-log1p-u 74.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(1 + x\right) \cdot \left(x \cdot \left(x - 1\right)\right)}\right)\right)} \]

   2. expm1-udef 58.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(1 + x\right) \cdot \left(x \cdot \left(x - 1\right)\right)}\right)} - 1} \]

   3. +-commutative 58.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(x + 1\right)} \cdot \left(x \cdot \left(x - 1\right)\right)}\right)} - 1 \]

   4. sub-neg 58.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(x \cdot \color{blue}{\left(x + \left(-1\right)\right)}\right)}\right)} - 1 \]

   5. metadata-eval 58.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(x \cdot \left(x + \color{blue}{-1}\right)\right)}\right)} - 1 \]

8. Applied egg-rr 58.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 74.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\right)\right)} \]

   2. expm1-log1p 99.7%

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

   3. associate-/r* 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 5: 83.8% accurate, 1.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.0) (- (* -2.0 x_m) (/ 2.0 x_m)) 0.0)))

C

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

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
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = ((-2.0d0) * x_m) - (2.0d0 / x_m)
    else
        tmp = 0.0d0
    end if
    code = x_s * tmp
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) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = (-2.0 * x_m) - (2.0 / x_m);
	} else {
		tmp = 0.0;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = (-2.0 * x_m) - (2.0 / x_m)
	else:
		tmp = 0.0
	return x_s * tmp

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = (-2.0 * x_m) - (2.0 / x_m);
	else
		tmp = 0.0;
	end
	tmp_2 = x_s * tmp;
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 * If[LessEqual[x$95$m, 1.0], N[(N[(-2.0 * x$95$m), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision], 0.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 88.6%

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

      1. sub-neg 88.6%

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

      2. distribute-neg-frac 88.6%

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

      3. metadata-eval 88.6%

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

      4. metadata-eval 88.6%

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

      5. metadata-eval 88.6%

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

      6. associate-/r* 88.6%

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

      7. metadata-eval 88.6%

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

      8. neg-mul-1 88.6%

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

      9. +-commutative 88.6%

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

      10. associate-+l+ 88.6%

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

      11. +-commutative 88.6%

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

      12. neg-mul-1 88.6%

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

      13. metadata-eval 88.6%

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

      14. associate-/r* 88.6%

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

      15. metadata-eval 88.6%

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

      16. metadata-eval 88.6%

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

      17. +-commutative 88.6%

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

      18. +-commutative 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. associate-*r/ 66.9%

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

      2. metadata-eval 66.9%

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

   6. Simplified 66.9%

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

   if 1 < x

   1. Initial program 67.5%

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

      1. associate-+l- 67.5%

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

      2. sub-neg 67.5%

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

      3. +-commutative 67.5%

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

      4. sub-neg 67.5%

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

      5. distribute-neg-in 67.5%

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

      6. distribute-neg-frac 67.5%

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

      7. metadata-eval 67.5%

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

      8. remove-double-neg 67.5%

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

      9. sub-neg 67.5%

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

      10. metadata-eval 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in x around inf 67.6%

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

   5. Taylor expanded in x around inf 67.2%

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

   6. Taylor expanded in x around 0 67.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 83.6% accurate, 3.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.0) (/ -2.0 x_m) 0.0)))

C

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

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
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = (-2.0d0) / x_m
    else
        tmp = 0.0d0
    end if
    code = x_s * tmp
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) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = -2.0 / x_m;
	} else {
		tmp = 0.0;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = -2.0 / x_m
	else:
		tmp = 0.0
	return x_s * tmp

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = -2.0 / x_m;
	else
		tmp = 0.0;
	end
	tmp_2 = x_s * tmp;
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 * If[LessEqual[x$95$m, 1.0], N[(-2.0 / x$95$m), $MachinePrecision], 0.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 88.6%

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

      1. sub-neg 88.6%

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

      2. distribute-neg-frac 88.6%

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

      3. metadata-eval 88.6%

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

      4. metadata-eval 88.6%

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

      5. metadata-eval 88.6%

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

      6. associate-/r* 88.6%

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

      7. metadata-eval 88.6%

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

      8. neg-mul-1 88.6%

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

      9. +-commutative 88.6%

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

      10. associate-+l+ 88.6%

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

      11. +-commutative 88.6%

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

      12. neg-mul-1 88.6%

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

      13. metadata-eval 88.6%

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

      14. associate-/r* 88.6%

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

      15. metadata-eval 88.6%

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

      16. metadata-eval 88.6%

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

      17. +-commutative 88.6%

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

      18. +-commutative 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 67.2%

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

   if 1 < x

   1. Initial program 67.5%

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

      1. associate-+l- 67.5%

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

      2. sub-neg 67.5%

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

      3. +-commutative 67.5%

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

      4. sub-neg 67.5%

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

      5. distribute-neg-in 67.5%

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

      6. distribute-neg-frac 67.5%

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

      7. metadata-eval 67.5%

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

      8. remove-double-neg 67.5%

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

      9. sub-neg 67.5%

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

      10. metadata-eval 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in x around inf 67.6%

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

   5. Taylor expanded in x around inf 67.2%

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

   6. Taylor expanded in x around 0 67.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 34.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.0;
}

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 * 0.0d0
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 * 0.0;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.0

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.0)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. associate-+l- 83.8%

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

   2. sub-neg 83.8%

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

   3. +-commutative 83.8%

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

   4. sub-neg 83.8%

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

   5. distribute-neg-in 83.8%

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

   6. distribute-neg-frac 83.8%

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

   7. metadata-eval 83.8%

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

   8. remove-double-neg 83.8%

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

   9. sub-neg 83.8%

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

   10. metadata-eval 83.8%

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

3. Simplified 83.8%

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

4. Taylor expanded in x around inf 41.2%

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

5. Taylor expanded in x around inf 32.7%

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

6. Taylor expanded in x around 0 32.7%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 32.7%

   \[\leadsto 0 \]

Developer target: 99.6% 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 2023332 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

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