3frac (problem 3.3.3)

Percentage Accurate: 68.5% → 99.8%

Time: 7.9s

Alternatives: 5

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: 68.5% 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 71.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-sub N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-lft-identity N/A

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

   14. lower--.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 23.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. *-lft-identity N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

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: 99.1% accurate, 1.8× speedup

Math

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

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

Derivation

1. Initial program 71.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-sub N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-lft-identity N/A

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

   14. lower--.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 23.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-lft-identity N/A

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

   11. distribute-rgt-in N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lift--.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

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 (* (fma x_m 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 / (fma(x_m, 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(fma(x_m, x_m, -1.0) * x_m)))
end

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-sub N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-lft-identity N/A

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

   14. lower--.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 23.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. difference-of-sqr-1 N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. lift-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 99.1

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

9. Applied rewrites 99.1%

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

Alternative 4: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. lower-pow.f64 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.7%

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

Alternative 5: 5.0% accurate, 3.8× speedup

Math

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

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 71.2%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.0

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

5. Applied rewrites 5.0%

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

Developer Target 1: 99.1% 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 2024249 
(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))))