3frac (problem 3.3.3)

Percentage Accurate: 69.7% → 99.8%

Time: 9.4s

Alternatives: 5

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: 69.7% 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

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

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 71.4%

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

   1. associate-+l- 71.4%

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

   2. +-commutative 71.4%

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

   3. sub-neg 71.4%

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

   4. metadata-eval 71.4%

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

3. Simplified 71.4%

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

   1. frac-sub 19.3%

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

   2. associate-/r* 71.4%

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

   3. *-rgt-identity 71.4%

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

   4. +-commutative 71.4%

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

   5. distribute-lft-in 71.4%

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

   6. metadata-eval 71.4%

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

6. Applied egg-rr 71.4%

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

   1. associate-/l/ 19.3%

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

   2. frac-sub 20.1%

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

   3. *-un-lft-identity 20.1%

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

   4. +-commutative 20.1%

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

   5. +-commutative 20.1%

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

   6. fma-define 20.1%

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

   7. +-commutative 20.1%

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

8. Applied egg-rr 20.1%

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

9. Taylor expanded in x around 0 98.2%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.8%

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

    3. *-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

    1. associate-*r/ 99.8%

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

    2. *-rgt-identity 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 99.1% 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 71.4%

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

   1. associate-+l- 71.4%

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

   2. +-commutative 71.4%

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

   3. sub-neg 71.4%

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

   4. metadata-eval 71.4%

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

3. Simplified 71.4%

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

   1. frac-sub 19.3%

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

   2. associate-/r* 71.4%

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

   3. *-rgt-identity 71.4%

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

   4. +-commutative 71.4%

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

   5. distribute-lft-in 71.4%

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

   6. metadata-eval 71.4%

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

6. Applied egg-rr 71.4%

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

   1. associate-/l/ 19.3%

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

   2. frac-sub 20.1%

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

   3. *-un-lft-identity 20.1%

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

   4. +-commutative 20.1%

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

   5. +-commutative 20.1%

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

   6. fma-define 20.1%

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

   7. +-commutative 20.1%

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

8. Applied egg-rr 20.1%

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

9. Taylor expanded in x around 0 98.2%

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

10. Final simplification 98.2%

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

Alternative 3: 68.2% 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 1 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 71.4%

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

   1. associate-+l- 71.4%

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

   2. +-commutative 71.4%

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

   3. sub-neg 71.4%

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

   4. metadata-eval 71.4%

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

3. Simplified 71.4%

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

5. Taylor expanded in x around inf 69.3%

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

6. Final simplification 69.3%

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

Alternative 4: 5.1% accurate, 5.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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 71.4%

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

   1. associate-+l- 71.4%

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

   2. +-commutative 71.4%

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

   3. sub-neg 71.4%

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

   4. metadata-eval 71.4%

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

3. Simplified 71.4%

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

5. Taylor expanded in x around 0 5.0%

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

6. Final simplification 5.0%

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

Alternative 5: 5.1% accurate, 5.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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 71.4%

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

   1. associate-+l- 71.4%

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

   2. +-commutative 71.4%

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

   3. sub-neg 71.4%

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

   4. metadata-eval 71.4%

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

3. Simplified 71.4%

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

5. Taylor expanded in x around inf 69.3%

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

6. Taylor expanded in x around 0 5.1%

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

7. Final simplification 5.1%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(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))))