3frac (problem 3.3.3)

Percentage Accurate: 68.8% → 99.8%

Time: 11.1s

Alternatives: 8

Speedup: 2.1×

Specification

\[\left|x\right| > 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% 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

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

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 (fma x_m x_m x_m)) (+ x_m -1.0))))

C

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

Julia

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

Derivation

1. Initial program 67.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 18.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.8%

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

   1. lift-approx N/A

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

   2. lift-+.f64 N/A

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

   3. *-lft-identity N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

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

Derivation

1. Initial program 67.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 18.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.8%

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

   1. lift-approx N/A

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

   2. lift-+.f64 N/A

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

   3. *-lft-identity N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 99.8%

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

Alternative 3: 98.7% accurate, 1.6× speedup

Math

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

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 x)) 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 * x)) / 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 * x)) / 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 * x)) / 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 * x)) / 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 * x)) / 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 * x)) / 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 * x), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 18.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.8%

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

   1. lift-approx N/A

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

   2. lift-+.f64 N/A

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

   3. *-lft-identity N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. associate-*r* N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around inf

    \[\leadsto \frac{\color{blue}{\frac{2}{{x}^{2}}}}{x} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 99.5

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

12. Applied rewrites 99.5%

    \[\leadsto \frac{\color{blue}{\frac{2}{x \cdot x}}}{x} \]
13. Add Preprocessing

Alternative 4: 99.2% accurate, 1.8× speedup

Math

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

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 (* (fma x_m x_m x_m) (+ x_m -1.0)))))

C

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

Julia

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

Derivation

1. Initial program 67.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 18.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.8%

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

   1. lift-+.f64 N/A

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

   2. *-lft-identity N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-lft-identity N/A

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

   8. +-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 98.8

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

9. Applied rewrites 98.8%

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

Alternative 5: 99.2% accurate, 2.0× speedup

Math

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

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 (fma x x -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 * fma(x, x, -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(x * fma(x, x, -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[(x * N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 18.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. *-lft-identity N/A

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

   3. lft-mult-inverse N/A

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

   4. distribute-lft-neg-out N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. distribute-rgt-in N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. associate-*l* N/A

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

   14. unpow2 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

10. Applied rewrites 98.8%

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

Alternative 6: 98.1% accurate, 2.1× speedup

Math

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

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 (* x x)))))

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 * (x * x)));
}

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 * (x * x)))
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 * (x * x)));
}

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 * (x * x)))

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 * Float64(x * x))))
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 * (x * x)));
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 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\frac{2}{x \cdot \left(x \cdot x\right)}} \]
6. Add Preprocessing

Alternative 7: 51.9% accurate, 2.6× speedup

Math

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

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 (fma x_m x_m 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 / fma(x_m, x_m, 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 / fma(x_m, x_m, 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 * x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 18.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.8%

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

   1. lift-approx N/A

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

   2. lift-+.f64 N/A

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

   3. *-lft-identity N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 53.5%

    \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, x\right)} \]
13. Add Preprocessing

Alternative 8: 5.0% accurate, 3.8× speedup

Math

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

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)))

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);
}

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)
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);
}

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)

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))
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);
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), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.1

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

5. Applied rewrites 5.1%

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

Developer Target 1: 99.2% accurate, 1.8× 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 2024212 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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