3frac (problem 3.3.3)

Percentage Accurate: 70.2% → 99.8%

Time: 8.4s

Alternatives: 6

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. frac-sub 20.9%

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

   3. frac-add 21.8%

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

   4. *-un-lft-identity 21.8%

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

   5. *-commutative 21.8%

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

   6. fma-define 19.8%

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

   7. +-commutative 19.8%

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

   8. sub-neg 19.8%

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

   9. metadata-eval 19.8%

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

   10. /-rgt-identity 19.8%

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

   11. *-un-lft-identity 19.8%

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

   12. /-rgt-identity 19.8%

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

   13. +-commutative 19.8%

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

   14. sub-neg 19.8%

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

   15. metadata-eval 19.8%

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

4. Applied egg-rr 19.8%

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

5. Taylor expanded in x around 0 99.6%

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

   1. metadata-eval 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-/r* 99.8%

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

   4. sub-neg 99.8%

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

   5. metadata-eval 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. *-un-lft-identity 99.8%

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

   8. unpow2 99.8%

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

7. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

   2. distribute-rgt1-in 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. frac-sub 20.9%

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

   3. frac-add 21.8%

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

   4. *-un-lft-identity 21.8%

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

   5. *-commutative 21.8%

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

   6. fma-define 19.8%

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

   7. +-commutative 19.8%

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

   8. sub-neg 19.8%

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

   9. metadata-eval 19.8%

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

   10. /-rgt-identity 19.8%

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

   11. *-un-lft-identity 19.8%

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

   12. /-rgt-identity 19.8%

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

   13. +-commutative 19.8%

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

   14. sub-neg 19.8%

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

   15. metadata-eval 19.8%

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

4. Applied egg-rr 19.8%

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

5. Taylor expanded in x around 0 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 69.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in x around inf 61.4%

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

4. Final simplification 61.4%

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

Alternative 4: 54.7% accurate, 2.1× speedup

Math

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

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 64.3%

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

   1. +-commutative 64.3%

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

   2. frac-sub 20.9%

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

   3. frac-add 21.8%

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

   4. *-un-lft-identity 21.8%

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

   5. *-commutative 21.8%

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

   6. fma-define 19.8%

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

   7. +-commutative 19.8%

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

   8. sub-neg 19.8%

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

   9. metadata-eval 19.8%

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

   10. /-rgt-identity 19.8%

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

   11. *-un-lft-identity 19.8%

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

   12. /-rgt-identity 19.8%

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

   13. +-commutative 19.8%

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

   14. sub-neg 19.8%

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

   15. metadata-eval 19.8%

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

4. Applied egg-rr 19.8%

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

5. Taylor expanded in x around 0 99.6%

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

6. Taylor expanded in x around 0 47.6%

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

7. Final simplification 47.6%

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

Alternative 5: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in x around inf 61.4%

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

4. Taylor expanded in x around 0 4.7%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
5. Add Preprocessing

Alternative 6: 5.1% accurate, 5.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 64.3%

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

3. Taylor expanded in x around 0 4.6%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (/ 2.0 (* x (- (* x x) 1.0)))

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