3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.2%

Time: 10.6s

Alternatives: 7

Speedup: 1.7×

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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(1 - x_m\right) \cdot \left(-1 - x_m\right)\\ x_s \cdot \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x_m + 1} - \frac{2}{x_m}\right) + \frac{1}{x_m + -1} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{-2}{x_m}}{x_m \cdot \left(1 - x_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m \cdot \left(2 \cdot x_m\right) + -2 \cdot t_0}{x_m \cdot t_0}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (- 1.0 x_m) (- -1.0 x_m))))
   (*
    x_s
    (if (<= (+ (- (/ 1.0 (+ x_m 1.0)) (/ 2.0 x_m)) (/ 1.0 (+ x_m -1.0))) 5e-26)
      (/ (/ -2.0 x_m) (* x_m (- 1.0 x_m)))
      (/ (+ (* x_m (* 2.0 x_m)) (* -2.0 t_0)) (* x_m t_0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (1.0 - x_m) * (-1.0 - x_m);
	double tmp;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-26) {
		tmp = (-2.0 / x_m) / (x_m * (1.0 - x_m));
	} else {
		tmp = ((x_m * (2.0 * x_m)) + (-2.0 * t_0)) / (x_m * t_0);
	}
	return x_s * tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x_m) * ((-1.0d0) - x_m)
    if ((((1.0d0 / (x_m + 1.0d0)) - (2.0d0 / x_m)) + (1.0d0 / (x_m + (-1.0d0)))) <= 5d-26) then
        tmp = ((-2.0d0) / x_m) / (x_m * (1.0d0 - x_m))
    else
        tmp = ((x_m * (2.0d0 * x_m)) + ((-2.0d0) * t_0)) / (x_m * t_0)
    end if
    code = x_s * tmp
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) {
	double t_0 = (1.0 - x_m) * (-1.0 - x_m);
	double tmp;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-26) {
		tmp = (-2.0 / x_m) / (x_m * (1.0 - x_m));
	} else {
		tmp = ((x_m * (2.0 * x_m)) + (-2.0 * t_0)) / (x_m * t_0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = (1.0 - x_m) * (-1.0 - x_m)
	tmp = 0
	if (((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-26:
		tmp = (-2.0 / x_m) / (x_m * (1.0 - x_m))
	else:
		tmp = ((x_m * (2.0 * x_m)) + (-2.0 * t_0)) / (x_m * t_0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(1.0 - x_m) * Float64(-1.0 - x_m))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 / Float64(x_m + 1.0)) - Float64(2.0 / x_m)) + Float64(1.0 / Float64(x_m + -1.0))) <= 5e-26)
		tmp = Float64(Float64(-2.0 / x_m) / Float64(x_m * Float64(1.0 - x_m)));
	else
		tmp = Float64(Float64(Float64(x_m * Float64(2.0 * x_m)) + Float64(-2.0 * t_0)) / Float64(x_m * t_0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = (1.0 - x_m) * (-1.0 - x_m);
	tmp = 0.0;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-26)
		tmp = (-2.0 / x_m) / (x_m * (1.0 - x_m));
	else
		tmp = ((x_m * (2.0 * x_m)) + (-2.0 * t_0)) / (x_m * t_0);
	end
	tmp_2 = x_s * tmp;
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_] := Block[{t$95$0 = N[(N[(1.0 - x$95$m), $MachinePrecision] * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-26], N[(N[(-2.0 / x$95$m), $MachinePrecision] / N[(x$95$m * N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(2.0 * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 5.00000000000000019e-26

   1. Initial program 73.7%

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

      1. frac-sub 21.4%

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

      2. associate-/r* 73.6%

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

      3. /-rgt-identity 73.6%

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

      4. *-un-lft-identity 73.6%

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

      5. /-rgt-identity 73.6%

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

      6. +-commutative 73.6%

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

      7. +-commutative 73.6%

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

   4. Applied egg-rr 73.6%

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

      1. frac-2neg 73.6%

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

      2. metadata-eval 73.6%

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

      3. frac-add 73.6%

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

      4. *-commutative 73.6%

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

      5. +-commutative 73.6%

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

      6. +-commutative 73.6%

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

      7. sub-neg 73.6%

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

      8. metadata-eval 73.6%

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

      9. sub-neg 73.6%

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

      10. metadata-eval 73.6%

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

   6. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. mul-1-neg 73.6%

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

      3. unsub-neg 73.6%

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

      4. *-commutative 73.6%

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

      5. neg-sub0 73.6%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(x + -1\right)\right)} \cdot \frac{x - 2 \cdot \left(x + 1\right)}{x + 1} - x}{x \cdot \left(-\left(x + -1\right)\right)} \]

      6. +-commutative 73.6%

         \[\leadsto \frac{\left(0 - \color{blue}{\left(-1 + x\right)}\right) \cdot \frac{x - 2 \cdot \left(x + 1\right)}{x + 1} - x}{x \cdot \left(-\left(x + -1\right)\right)} \]

      7. associate--r+ 73.6%

         \[\leadsto \frac{\color{blue}{\left(\left(0 - -1\right) - x\right)} \cdot \frac{x - 2 \cdot \left(x + 1\right)}{x + 1} - x}{x \cdot \left(-\left(x + -1\right)\right)} \]

      8. metadata-eval 73.6%

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

      9. div-sub 73.6%

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

      10. associate-*r/ 73.6%

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

      11. *-inverses 73.6%

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

      12. metadata-eval 73.6%

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

      13. sub-neg 73.6%

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

      14. metadata-eval 73.6%

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

      15. neg-sub0 73.6%

          \[\leadsto \frac{\left(1 - x\right) \cdot \left(\frac{x}{x + 1} + -2\right) - x}{x \cdot \color{blue}{\left(0 - \left(x + -1\right)\right)}} \]

      16. +-commutative 73.6%

          \[\leadsto \frac{\left(1 - x\right) \cdot \left(\frac{x}{x + 1} + -2\right) - x}{x \cdot \left(0 - \color{blue}{\left(-1 + x\right)}\right)} \]

   8. Simplified 73.6%

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

   9. Taylor expanded in x around inf 99.2%

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

       1. associate-*r/ 99.2%

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

       2. metadata-eval 99.2%

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

       3. associate-*r/ 99.2%

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

       4. metadata-eval 99.2%

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

   11. Simplified 99.2%

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

   12. Taylor expanded in x around inf 98.2%

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

   if 5.00000000000000019e-26 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 62.6%

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

      1. sub-neg 62.6%

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

      2. distribute-neg-frac 62.6%

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

      3. metadata-eval 62.6%

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

      4. metadata-eval 62.6%

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

      5. metadata-eval 62.6%

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

      6. associate-/r* 62.6%

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

      7. metadata-eval 62.6%

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

      8. neg-mul-1 62.6%

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

      9. +-commutative 62.6%

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

      10. associate-+l+ 61.7%

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

      11. +-commutative 61.7%

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

      12. neg-mul-1 61.7%

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

      13. metadata-eval 61.7%

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

      14. associate-/r* 61.7%

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

      15. metadata-eval 61.7%

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

      16. metadata-eval 61.7%

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

      17. +-commutative 61.7%

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

      18. +-commutative 61.7%

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

      19. sub-neg 61.7%

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

      20. metadata-eval 61.7%

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

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{-2}{x} + \left(\frac{1}{1 + x} + \frac{1}{x + -1}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 61.7%

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

      2. frac-2neg 61.7%

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

      3. metadata-eval 61.7%

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

      4. frac-2neg 61.7%

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

      5. metadata-eval 61.7%

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

      6. frac-add 62.1%

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

      7. +-commutative 62.1%

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

      8. distribute-neg-in 62.1%

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

      9. metadata-eval 62.1%

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

      10. sub-neg 62.1%

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

      11. +-commutative 62.1%

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

      12. distribute-neg-in 62.1%

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

      13. metadata-eval 62.1%

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

      14. +-commutative 62.1%

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

      15. distribute-neg-in 62.1%

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

      16. metadata-eval 62.1%

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

      17. +-commutative 62.1%

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

      18. distribute-neg-in 62.1%

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

   6. Applied egg-rr 62.1%

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

      1. +-commutative 62.1%

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

      2. *-commutative 62.1%

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

      3. *-commutative 62.1%

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

      4. *-commutative 62.1%

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

      5. associate-/r* 61.7%

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

   8. Simplified 61.7%

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

      1. +-commutative 61.7%

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

      2. associate-/l/ 62.1%

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

      3. frac-add 99.6%

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

      4. count-2 99.6%

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

   10. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\frac{\left(2 \cdot x\right) \cdot x + \left(\left(-1 - x\right) \cdot \left(1 - x\right)\right) \cdot -2}{\left(\left(-1 - x\right) \cdot \left(1 - x\right)\right) \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 \cdot x\right) + -2 \cdot \left(\left(1 - x\right) \cdot \left(-1 - x\right)\right)}{x \cdot \left(\left(1 - x\right) \cdot \left(-1 - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(2 \cdot {x_m}^{-3}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* 2.0 (pow x_m -3.0))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (2.0 * pow(x_m, -3.0));
}

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 ** (-3.0d0)))
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 * Math.pow(x_m, -3.0));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (2.0 * math.pow(x_m, -3.0))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(2.0 * (x_m ^ -3.0)))
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 ^ -3.0));
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[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in x around inf 98.1%

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

   1. expm1-log1p-u 98.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   2. expm1-udef 72.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1} \]

   3. div-inv 72.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1 \]

   4. pow-flip 72.3%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1 \]

   5. metadata-eval 72.3%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1 \]

5. Applied egg-rr 72.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 98.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} \]

   2. expm1-log1p 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

   \[\leadsto 2 \cdot {x}^{-3} \]
9. Add Preprocessing

Alternative 3: 68.2% accurate, 1.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (/ 1.0 (+ x_m -1.0)) (/ -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 + -1.0)) + (-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 + (-1.0d0))) + ((-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 + -1.0)) + (-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 + -1.0)) + (-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(1.0 / Float64(x_m + -1.0)) + 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 + -1.0)) + (-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[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in x around inf 71.8%

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

4. Final simplification 71.8%

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

Alternative 4: 97.8% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.5%

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

   1. frac-sub 22.0%

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

   2. associate-/r* 73.4%

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

   3. /-rgt-identity 73.4%

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

   4. *-un-lft-identity 73.4%

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

   5. /-rgt-identity 73.4%

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

   6. +-commutative 73.4%

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

   7. +-commutative 73.4%

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

4. Applied egg-rr 73.4%

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

   1. frac-2neg 73.4%

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

   2. metadata-eval 73.4%

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

   3. frac-add 73.5%

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

   4. *-commutative 73.5%

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

   5. +-commutative 73.5%

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

   6. +-commutative 73.5%

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

   7. sub-neg 73.5%

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

   8. metadata-eval 73.5%

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

   9. sub-neg 73.5%

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

   10. metadata-eval 73.5%

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

6. Applied egg-rr 73.5%

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

   1. *-commutative 73.5%

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

   2. mul-1-neg 73.5%

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

   3. unsub-neg 73.5%

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

   4. *-commutative 73.5%

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

   5. neg-sub0 73.5%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x + -1\right)\right)} \cdot \frac{x - 2 \cdot \left(x + 1\right)}{x + 1} - x}{x \cdot \left(-\left(x + -1\right)\right)} \]

   6. +-commutative 73.5%

      \[\leadsto \frac{\left(0 - \color{blue}{\left(-1 + x\right)}\right) \cdot \frac{x - 2 \cdot \left(x + 1\right)}{x + 1} - x}{x \cdot \left(-\left(x + -1\right)\right)} \]

   7. associate--r+ 73.5%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - -1\right) - x\right)} \cdot \frac{x - 2 \cdot \left(x + 1\right)}{x + 1} - x}{x \cdot \left(-\left(x + -1\right)\right)} \]

   8. metadata-eval 73.5%

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

   9. div-sub 73.4%

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

   10. associate-*r/ 73.4%

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

   11. *-inverses 73.4%

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

   12. metadata-eval 73.4%

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

   13. sub-neg 73.4%

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

   14. metadata-eval 73.4%

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

   15. neg-sub0 73.4%

       \[\leadsto \frac{\left(1 - x\right) \cdot \left(\frac{x}{x + 1} + -2\right) - x}{x \cdot \color{blue}{\left(0 - \left(x + -1\right)\right)}} \]

   16. +-commutative 73.4%

       \[\leadsto \frac{\left(1 - x\right) \cdot \left(\frac{x}{x + 1} + -2\right) - x}{x \cdot \left(0 - \color{blue}{\left(-1 + x\right)}\right)} \]

8. Simplified 73.4%

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

9. Taylor expanded in x around inf 98.5%

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

    1. associate-*r/ 98.5%

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

    2. metadata-eval 98.5%

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

    3. associate-*r/ 98.5%

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

    4. metadata-eval 98.5%

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

11. Simplified 98.5%

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

12. Taylor expanded in x around inf 97.3%

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

13. Final simplification 97.3%

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

Alternative 5: 67.8% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.5%

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

   1. sub-neg 73.5%

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

   2. distribute-neg-frac 73.5%

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

   3. metadata-eval 73.5%

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

   4. metadata-eval 73.5%

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

   5. metadata-eval 73.5%

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

   6. associate-/r* 73.5%

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

   7. metadata-eval 73.5%

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

   8. neg-mul-1 73.5%

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

   9. +-commutative 73.5%

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

   10. associate-+l+ 73.4%

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

   11. +-commutative 73.4%

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

   12. neg-mul-1 73.4%

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

   13. metadata-eval 73.4%

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

   14. associate-/r* 73.4%

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

   15. metadata-eval 73.4%

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

   16. metadata-eval 73.4%

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

   17. +-commutative 73.4%

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

   18. +-commutative 73.4%

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

   19. sub-neg 73.4%

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

   20. metadata-eval 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in x around inf 71.6%

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

6. Final simplification 71.6%

   \[\leadsto \frac{2}{x} + \frac{-2}{x} \]
7. Add Preprocessing

Alternative 6: 5.0% 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 1 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 73.5%

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

3. Taylor expanded in x around 0 5.1%

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

4. Final simplification 5.1%

   \[\leadsto \frac{-2}{x} \]
5. Add Preprocessing

Alternative 7: 3.9% accurate, 15.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 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;
}

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
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;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 1.0

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 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;
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 * 1.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in x around 0 3.4%

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

   1. associate-*r/ 3.4%

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

   2. metadata-eval 3.4%

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

5. Simplified 3.4%

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

6. Taylor expanded in x around inf 3.4%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 3.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 99.2% 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 2024021 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

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