3frac (problem 3.3.3)

Percentage Accurate: 68.4% → 99.8%

Time: 10.9s

Alternatives: 6

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: 68.4% 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, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 66.2%

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

   1. frac-sub N/A

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

   2. div-inv N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lft-identity N/A

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

   5. --lowering--.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. +-commutative N/A

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

   13. +-lowering-+.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-lowering-+.f64 7.3

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

4. Applied egg-rr 7.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. frac-add N/A

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

   4. /-rgt-identity N/A

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

   5. /-lowering-/.f64 N/A

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

6. Applied egg-rr 18.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 99.4%

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

    1. *-commutative N/A

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

    2. associate-/r* N/A

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

    3. distribute-lft1-in N/A

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

    4. metadata-eval N/A

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

    5. sub-neg N/A

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

    6. associate-/r* N/A

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

    7. /-lowering-/.f64 N/A

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

    8. /-lowering-/.f64 N/A

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

    9. /-lowering-/.f64 N/A

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

    10. +-lowering-+.f64 N/A

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

    11. sub-neg N/A

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

    12. metadata-eval N/A

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

    13. metadata-eval N/A

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

    14. +-lowering-+.f64 N/A

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

    15. metadata-eval 99.8

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

11. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{x + -1}}{x + 1}}{x}} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

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 66.2%

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

   1. frac-sub N/A

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

   2. div-inv N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lft-identity N/A

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

   5. --lowering--.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. +-commutative N/A

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

   13. +-lowering-+.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-lowering-+.f64 7.3

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

4. Applied egg-rr 7.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. frac-add N/A

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

   4. /-rgt-identity N/A

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

   5. /-lowering-/.f64 N/A

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

6. Applied egg-rr 18.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 99.4%

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

    1. associate-/r* N/A

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

    2. /-lowering-/.f64 N/A

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

    3. /-lowering-/.f64 N/A

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

    4. accelerator-lowering-fma.f64 N/A

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

    5. +-lowering-+.f64 99.8

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

11. Applied egg-rr 99.8%

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

Alternative 3: 99.2% accurate, 2.0× speedup

Math

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

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 (fma x_m x_m -1.0)))))

C

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

   1. frac-sub N/A

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

   2. div-inv N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lft-identity N/A

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

   5. --lowering--.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. +-commutative N/A

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

   13. +-lowering-+.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-lowering-+.f64 7.3

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

4. Applied egg-rr 7.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. frac-add N/A

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

   4. /-rgt-identity N/A

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

   5. /-lowering-/.f64 N/A

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

6. Applied egg-rr 18.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 99.4%

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

    1. distribute-lft1-in N/A

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

    2. metadata-eval N/A

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

    3. sub-neg N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

    8. metadata-eval N/A

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

    9. metadata-eval N/A

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

    10. sub-neg N/A

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

    11. metadata-eval N/A

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

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

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

    13. metadata-eval N/A

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

    14. *-lowering-*.f64 N/A

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

    15. sub-neg N/A

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

    16. metadata-eval N/A

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

    17. metadata-eval N/A

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

    18. accelerator-lowering-fma.f64 99.4

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

11. Applied egg-rr 99.4%

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

Alternative 4: 98.3% accurate, 2.1× speedup

Math

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

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 (* 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(x_m * Float64(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[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.f64 98.4

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

5. Simplified 98.4%

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

Alternative 5: 53.6% accurate, 2.3× speedup

Math

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

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)))))

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)));
}

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))))
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)));
}

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)))

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(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)));
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[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

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

   1. frac-sub N/A

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

   2. div-inv N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lft-identity N/A

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

   5. --lowering--.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. +-commutative N/A

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

   13. +-lowering-+.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-lowering-+.f64 7.3

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

4. Applied egg-rr 7.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. frac-add N/A

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

   4. /-rgt-identity N/A

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

   5. /-lowering-/.f64 N/A

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

6. Applied egg-rr 18.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 99.4%

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

10. Taylor expanded in x around 0

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

12. Simplified 50.6%

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

Alternative 6: 5.1% 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 66.2%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 5.0

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

5. Simplified 5.0%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
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 2024198 
(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))))