3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.1%

Time: 9.4s

Alternatives: 7

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: 69.3% 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: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{\frac{1 - x\_m}{\left(0.5 \cdot x\_m\right) \cdot x\_m} - -2}{x\_m}}{\left(x\_m - 1\right) \cdot x\_m} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (/ (- (/ (- 1.0 x_m) (* (* 0.5 x_m) x_m)) -2.0) 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 * (((((1.0 - x_m) / ((0.5 * x_m) * x_m)) - -2.0) / x_m) / ((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 * (((((1.0d0 - x_m) / ((0.5d0 * x_m) * x_m)) - (-2.0d0)) / x_m) / ((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 * (((((1.0 - x_m) / ((0.5 * x_m) * x_m)) - -2.0) / x_m) / ((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 * (((((1.0 - x_m) / ((0.5 * x_m) * x_m)) - -2.0) / 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(Float64(Float64(Float64(1.0 - x_m) / Float64(Float64(0.5 * x_m) * x_m)) - -2.0) / x_m) / Float64(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 * (((((1.0 - x_m) / ((0.5 * x_m) * x_m)) - -2.0) / x_m) / ((x_m - 1.0) * x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(N[(1.0 - x$95$m), $MachinePrecision] / N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(N[(x$95$m - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. div-sub N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 99.2

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\frac{1 - x}{x \cdot \left(x \cdot 0.5\right)} - -2}{x}}{x \cdot \left(x - 1\right)} \]

10. Final simplification 99.2%

    \[\leadsto \frac{\frac{\frac{1 - x}{\left(0.5 \cdot x\right) \cdot x} - -2}{x}}{\left(x - 1\right) \cdot x} \]
11. Add Preprocessing

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (/ (- 2.0 (/ 2.0 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 - (2.0 / x_m)) / x_m) / ((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 - (2.0d0 / x_m)) / x_m) / ((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 - (2.0 / x_m)) / x_m) / ((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 - (2.0 / 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(Float64(2.0 - Float64(2.0 / x_m)) / x_m) / Float64(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 - (2.0 / x_m)) / x_m) / ((x_m - 1.0) * x_m));
end

Wolfram

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

Derivation

1. Initial program 69.1%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 98.8

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 3: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 98.4

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.8%

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

Alternative 4: 98.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Derivation

1. Initial program 69.1%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 98.4

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.7%

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

Alternative 5: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 98.4

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.4%

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

Alternative 6: 54.6% accurate, 2.7× speedup

Math

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

FPCore

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

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));
}

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))
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));
}

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))

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 * 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));
end

Wolfram

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

Derivation

1. Initial program 69.1%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 53.8%

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

8. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 53.8

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

10. Applied rewrites 53.8%

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

Alternative 7: 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 69.1%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.0

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

5. Applied rewrites 5.0%

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

Developer Target 1: 99.2% 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 2024254 
(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))))