3frac (problem 3.3.3)

Percentage Accurate: 69.7% → 99.8%

Time: 11.2s

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: 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

   1. frac-add 14.6%

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

   2. frac-add 13.6%

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

   3. *-un-lft-identity 13.6%

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

   4. fma-define 13.0%

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

   5. *-rgt-identity 13.0%

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

   6. +-commutative 13.0%

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

   7. distribute-rgt-in 13.0%

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

   8. metadata-eval 13.0%

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

   9. metadata-eval 13.0%

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

   10. fma-define 13.0%

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

   11. metadata-eval 13.0%

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

   12. distribute-rgt-in 13.0%

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

   13. neg-mul-1 13.0%

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

   14. sub-neg 13.0%

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

   15. pow2 13.0%

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

6. Applied egg-rr 13.0%

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

   1. +-commutative 13.0%

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

   2. sub-neg 13.0%

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

   3. +-commutative 13.0%

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

   4. unpow2 13.0%

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

   5. neg-mul-1 13.0%

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

   6. distribute-rgt-in 13.0%

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

8. Simplified 13.0%

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

9. Taylor expanded in x around 0 99.2%

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

    1. associate-/r* 99.7%

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

    2. div-inv 99.7%

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

    3. distribute-rgt-in 99.7%

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

    4. neg-mul-1 99.7%

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

    5. sub-neg 99.7%

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

    6. pow2 99.7%

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

11. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. associate-*r/ 99.8%

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

    3. metadata-eval 99.8%

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

    4. sub-neg 99.8%

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

    5. unpow2 99.8%

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

    6. neg-mul-1 99.8%

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

    7. distribute-rgt-in 99.8%

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

13. Simplified 99.8%

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

Alternative 2: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

   1. frac-add 14.6%

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

   2. frac-add 13.6%

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

   3. *-un-lft-identity 13.6%

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

   4. fma-define 13.0%

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

   5. *-rgt-identity 13.0%

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

   6. +-commutative 13.0%

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

   7. distribute-rgt-in 13.0%

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

   8. metadata-eval 13.0%

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

   9. metadata-eval 13.0%

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

   10. fma-define 13.0%

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

   11. metadata-eval 13.0%

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

   12. distribute-rgt-in 13.0%

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

   13. neg-mul-1 13.0%

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

   14. sub-neg 13.0%

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

   15. pow2 13.0%

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

6. Applied egg-rr 13.0%

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

   1. +-commutative 13.0%

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

   2. sub-neg 13.0%

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

   3. +-commutative 13.0%

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

   4. unpow2 13.0%

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

   5. neg-mul-1 13.0%

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

   6. distribute-rgt-in 13.0%

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

8. Simplified 13.0%

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

9. Taylor expanded in x around 0 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 68.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.3%

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

6. Final simplification 64.3%

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

Alternative 4: 52.3% accurate, 2.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

   1. frac-add 14.6%

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

   2. frac-add 13.6%

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

   3. *-un-lft-identity 13.6%

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

   4. fma-define 13.0%

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

   5. *-rgt-identity 13.0%

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

   6. +-commutative 13.0%

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

   7. distribute-rgt-in 13.0%

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

   8. metadata-eval 13.0%

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

   9. metadata-eval 13.0%

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

   10. fma-define 13.0%

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

   11. metadata-eval 13.0%

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

   12. distribute-rgt-in 13.0%

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

   13. neg-mul-1 13.0%

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

   14. sub-neg 13.0%

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

   15. pow2 13.0%

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

6. Applied egg-rr 13.0%

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

   1. +-commutative 13.0%

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

   2. sub-neg 13.0%

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

   3. +-commutative 13.0%

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

   4. unpow2 13.0%

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

   5. neg-mul-1 13.0%

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

   6. distribute-rgt-in 13.0%

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

8. Simplified 13.0%

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

9. Taylor expanded in x around 0 99.2%

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

10. Taylor expanded in x around 0 55.6%

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

    1. neg-mul-1 55.6%

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

12. Simplified 55.6%

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

13. Taylor expanded in x around 0 55.6%

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

    1. sub-neg 55.6%

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

    2. mul-1-neg 55.6%

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

    3. rem-3cbrt-lft 55.6%

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

    4. unpow2 55.6%

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

    5. distribute-rgt-neg-in 55.6%

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

    6. *-commutative 55.6%

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

    7. metadata-eval 55.6%

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

    8. +-commutative 55.6%

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

    9. distribute-lft-neg-in 55.6%

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

    10. *-commutative 55.6%

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

    11. unpow2 55.6%

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

    12. rem-3cbrt-lft 55.6%

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

    13. unsub-neg 55.6%

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

15. Simplified 55.6%

    \[\leadsto \frac{2}{\color{blue}{x \cdot \left(-1 - x\right)}} \]
16. 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 #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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.3%

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

6. Taylor expanded in x around 0 5.2%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
7. 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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around 0 5.2%

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

Alternative 7: 3.9% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.3%

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

6. Taylor expanded in x around 0 3.6%

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

   1. div-sub 3.6%

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

   2. sub-neg 3.6%

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

   3. *-inverses 3.6%

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

   4. distribute-neg-frac 3.6%

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

   5. metadata-eval 3.6%

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

8. Simplified 3.6%

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

9. Taylor expanded in x around inf 3.6%

   \[\leadsto \color{blue}{1} \]
10. 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 2024101 
(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))))