3frac (problem 3.3.3)

Percentage Accurate: 69.0% → 99.8%

Time: 9.8s

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: 69.0% 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

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

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

Derivation

1. Initial program 64.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. *-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 17.9

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 17.9

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

4. Applied rewrites 17.9%

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

6. Applied rewrites 20.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lift-fma.f64 N/A

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

    5. *-rgt-identity N/A

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

    6. distribute-lft-in N/A

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

    7. +-commutative N/A

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

    8. associate-/l/ N/A

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 64.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. *-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 17.9

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 17.9

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

4. Applied rewrites 17.9%

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

6. Applied rewrites 20.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.5%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-fma.f64 N/A

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

    4. *-rgt-identity N/A

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

    5. distribute-lft-in N/A

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

    6. +-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. +-commutative N/A

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

    9. distribute-lft-in N/A

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

    10. *-rgt-identity N/A

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

    11. lift-fma.f64 99.5

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

    12. lift-+.f64 N/A

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

    13. +-commutative N/A

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

    14. lift-neg.f64 N/A

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

    15. sub-neg N/A

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

    16. lower--.f64 99.5

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

11. Applied rewrites 99.5%

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

Alternative 3: 99.1% accurate, 1.8× speedup

Math

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

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

Derivation

1. Initial program 64.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. *-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 17.9

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 17.9

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

4. Applied rewrites 17.9%

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

6. Applied rewrites 20.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. distribute-lft-out-- N/A

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

    4. *-rgt-identity N/A

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

    5. unpow2 N/A

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

    6. cube-mult N/A

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

    7. lower--.f64 N/A

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

    8. cube-mult N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 99.5

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

12. Applied rewrites 99.5%

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

Alternative 4: 97.9% 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 64.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.8

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

5. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{2}{x \cdot \left(x \cdot x\right)}} \]
6. 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 64.2%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 4.8

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

5. Applied rewrites 4.8%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
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 2024237 
(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))))