3frac (problem 3.3.3)

Percentage Accurate: 85.0% → 98.8%

Time: 7.2s

Alternatives: 8

Speedup: 1.0×

Specification

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: 85.0% 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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{1}{1 + x_m}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;\left(t_0 - \frac{2}{x_m}\right) + \frac{1}{x_m + -1} \leq -20:\\ \;\;\;\;t_0 + \frac{\left(1 - x_m\right) + x_m \cdot 0.5}{\left(1 - x_m\right) \cdot \left(x_m \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x_m}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x_m))))
   (*
    x_s
    (if (<= (+ (- t_0 (/ 2.0 x_m)) (/ 1.0 (+ x_m -1.0))) -20.0)
      (+ t_0 (/ (+ (- 1.0 x_m) (* x_m 0.5)) (* (- 1.0 x_m) (* x_m -0.5))))
      (/ 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) {
	double t_0 = 1.0 / (1.0 + x_m);
	double tmp;
	if (((t_0 - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= -20.0) {
		tmp = t_0 + (((1.0 - x_m) + (x_m * 0.5)) / ((1.0 - x_m) * (x_m * -0.5)));
	} else {
		tmp = 2.0 / pow(x_m, 3.0);
	}
	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 / (1.0d0 + x_m)
    if (((t_0 - (2.0d0 / x_m)) + (1.0d0 / (x_m + (-1.0d0)))) <= (-20.0d0)) then
        tmp = t_0 + (((1.0d0 - x_m) + (x_m * 0.5d0)) / ((1.0d0 - x_m) * (x_m * (-0.5d0))))
    else
        tmp = 2.0d0 / (x_m ** 3.0d0)
    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 / (1.0 + x_m);
	double tmp;
	if (((t_0 - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= -20.0) {
		tmp = t_0 + (((1.0 - x_m) + (x_m * 0.5)) / ((1.0 - x_m) * (x_m * -0.5)));
	} else {
		tmp = 2.0 / Math.pow(x_m, 3.0);
	}
	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 / (1.0 + x_m)
	tmp = 0
	if ((t_0 - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= -20.0:
		tmp = t_0 + (((1.0 - x_m) + (x_m * 0.5)) / ((1.0 - x_m) * (x_m * -0.5)))
	else:
		tmp = 2.0 / math.pow(x_m, 3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(1.0 / Float64(1.0 + x_m))
	tmp = 0.0
	if (Float64(Float64(t_0 - Float64(2.0 / x_m)) + Float64(1.0 / Float64(x_m + -1.0))) <= -20.0)
		tmp = Float64(t_0 + Float64(Float64(Float64(1.0 - x_m) + Float64(x_m * 0.5)) / Float64(Float64(1.0 - x_m) * Float64(x_m * -0.5))));
	else
		tmp = Float64(2.0 / (x_m ^ 3.0));
	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 / (1.0 + x_m);
	tmp = 0.0;
	if (((t_0 - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= -20.0)
		tmp = t_0 + (((1.0 - x_m) + (x_m * 0.5)) / ((1.0 - x_m) * (x_m * -0.5)));
	else
		tmp = 2.0 / (x_m ^ 3.0);
	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[(1.0 / N[(1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(t$95$0 - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -20.0], N[(t$95$0 + N[(N[(N[(1.0 - x$95$m), $MachinePrecision] + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x$95$m), $MachinePrecision] * N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[x$95$m, 3.0], $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))) < -20

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. remove-double-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. div-inv 100.0%

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

      11. metadata-eval 100.0%

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

      12. div-inv 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -20 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 83.4%

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

      1. associate-+l- 83.4%

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

      2. sub-neg 83.4%

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

      3. +-commutative 83.4%

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

      4. sub-neg 83.4%

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

      5. distribute-neg-in 83.4%

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

      6. distribute-neg-frac 83.4%

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

      7. metadata-eval 83.4%

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

      8. remove-double-neg 83.4%

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

      9. sub-neg 83.4%

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

      10. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x around inf 64.0%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -20:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\left(1 - x\right) + x \cdot 0.5}{\left(1 - x\right) \cdot \left(x \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 2: 85.0% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (- (/ 1.0 (+ 1.0 x_m)) (/ 2.0 x_m)) (/ 1.0 (+ x_m -1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

2. Final simplification 87.4%

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

Alternative 3: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.65:\\ \;\;\;\;x_m \cdot -2 - \frac{2}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x_m} + \frac{1}{x_m + -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.65)
    (- (* x_m -2.0) (/ 2.0 x_m))
    (+ (/ -1.0 x_m) (/ 1.0 (+ x_m -1.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.65) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (-1.0 / x_m) + (1.0 / (x_m + -1.0));
	}
	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) :: tmp
    if (x_m <= 0.65d0) then
        tmp = (x_m * (-2.0d0)) - (2.0d0 / x_m)
    else
        tmp = ((-1.0d0) / x_m) + (1.0d0 / (x_m + (-1.0d0)))
    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 tmp;
	if (x_m <= 0.65) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (-1.0 / x_m) + (1.0 / (x_m + -1.0));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.65:
		tmp = (x_m * -2.0) - (2.0 / x_m)
	else:
		tmp = (-1.0 / x_m) + (1.0 / (x_m + -1.0))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.65)
		tmp = Float64(Float64(x_m * -2.0) - Float64(2.0 / x_m));
	else
		tmp = Float64(Float64(-1.0 / x_m) + Float64(1.0 / Float64(x_m + -1.0)));
	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)
	tmp = 0.0;
	if (x_m <= 0.65)
		tmp = (x_m * -2.0) - (2.0 / x_m);
	else
		tmp = (-1.0 / x_m) + (1.0 / (x_m + -1.0));
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.65], N[(N[(x$95$m * -2.0), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x$95$m), $MachinePrecision] + N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.650000000000000022

   1. Initial program 91.2%

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

      1. associate-+l- 91.2%

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

      2. sub-neg 91.2%

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

      3. +-commutative 91.2%

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

      4. sub-neg 91.2%

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

      5. distribute-neg-in 91.2%

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

      6. distribute-neg-frac 91.2%

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

      7. metadata-eval 91.2%

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

      8. remove-double-neg 91.2%

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

      9. sub-neg 91.2%

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

      10. metadata-eval 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in x around 0 68.6%

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

      1. associate-*r/ 68.6%

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

      2. metadata-eval 68.6%

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

   6. Simplified 68.6%

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

   if 0.650000000000000022 < x

   1. Initial program 76.1%

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

      1. associate-+l- 76.1%

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

      2. sub-neg 76.1%

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

      3. +-commutative 76.1%

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

      4. sub-neg 76.1%

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

      5. distribute-neg-in 76.1%

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

      6. distribute-neg-frac 76.1%

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

      7. metadata-eval 76.1%

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

      8. remove-double-neg 76.1%

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

      9. sub-neg 76.1%

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

      10. metadata-eval 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around inf 76.2%

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

      1. expm1-log1p-u 76.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)\right)\right)} \]

      2. expm1-udef 76.1%

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

      3. associate-+r+ 76.1%

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

      4. *-un-lft-identity 76.1%

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

      5. div-inv 76.1%

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

      6. distribute-rgt-out 76.1%

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

      7. metadata-eval 76.1%

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

   6. Applied egg-rr 76.1%

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

      1. expm1-def 76.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot -1 + \frac{1}{x + -1}\right)\right)} \]

      2. expm1-log1p 76.2%

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

      3. associate-*l/ 76.2%

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

      4. metadata-eval 76.2%

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

   8. Simplified 76.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \]

Alternative 4: 83.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = -x_m - (2.0 / x_m);
	} else {
		tmp = (-1.0 / x_m) + (1.0 / x_m);
	}
	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) :: tmp
    if (x_m <= 1.0d0) then
        tmp = -x_m - (2.0d0 / x_m)
    else
        tmp = ((-1.0d0) / x_m) + (1.0d0 / x_m)
    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 tmp;
	if (x_m <= 1.0) {
		tmp = -x_m - (2.0 / x_m);
	} else {
		tmp = (-1.0 / x_m) + (1.0 / x_m);
	}
	return x_s * tmp;
}

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(-x_m) - Float64(2.0 / x_m));
	else
		tmp = Float64(Float64(-1.0 / x_m) + Float64(1.0 / x_m));
	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)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = -x_m - (2.0 / x_m);
	else
		tmp = (-1.0 / x_m) + (1.0 / x_m);
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], N[((-x$95$m) - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x$95$m), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 91.2%

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

      1. associate-+l- 91.2%

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

      2. sub-neg 91.2%

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

      3. +-commutative 91.2%

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

      4. sub-neg 91.2%

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

      5. distribute-neg-in 91.2%

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

      6. distribute-neg-frac 91.2%

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

      7. metadata-eval 91.2%

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

      8. remove-double-neg 91.2%

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

      9. sub-neg 91.2%

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

      10. metadata-eval 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in x around 0 68.6%

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

   5. Taylor expanded in x around 0 68.4%

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

      1. neg-mul-1 68.4%

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

      2. associate-*r/ 68.4%

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

      3. metadata-eval 68.4%

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

   7. Simplified 68.4%

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

   if 1 < x

   1. Initial program 76.1%

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

      1. associate-+l- 76.1%

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

      2. sub-neg 76.1%

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

      3. +-commutative 76.1%

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

      4. sub-neg 76.1%

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

      5. distribute-neg-in 76.1%

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

      6. distribute-neg-frac 76.1%

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

      7. metadata-eval 76.1%

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

      8. remove-double-neg 76.1%

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

      9. sub-neg 76.1%

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

      10. metadata-eval 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around inf 76.1%

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

   5. Taylor expanded in x around inf 76.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(-x\right) - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x}\\ \end{array} \]

Alternative 5: 83.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (-1.0 / x_m) + (1.0 / x_m);
	}
	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) :: tmp
    if (x_m <= 1.0d0) then
        tmp = (x_m * (-2.0d0)) - (2.0d0 / x_m)
    else
        tmp = ((-1.0d0) / x_m) + (1.0d0 / x_m)
    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 tmp;
	if (x_m <= 1.0) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (-1.0 / x_m) + (1.0 / x_m);
	}
	return x_s * tmp;
}

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(x_m * -2.0) - Float64(2.0 / x_m));
	else
		tmp = Float64(Float64(-1.0 / x_m) + Float64(1.0 / x_m));
	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)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = (x_m * -2.0) - (2.0 / x_m);
	else
		tmp = (-1.0 / x_m) + (1.0 / x_m);
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], N[(N[(x$95$m * -2.0), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x$95$m), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 91.2%

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

      1. associate-+l- 91.2%

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

      2. sub-neg 91.2%

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

      3. +-commutative 91.2%

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

      4. sub-neg 91.2%

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

      5. distribute-neg-in 91.2%

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

      6. distribute-neg-frac 91.2%

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

      7. metadata-eval 91.2%

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

      8. remove-double-neg 91.2%

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

      9. sub-neg 91.2%

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

      10. metadata-eval 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in x around 0 68.6%

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

      1. associate-*r/ 68.6%

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

      2. metadata-eval 68.6%

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

   6. Simplified 68.6%

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

   if 1 < x

   1. Initial program 76.1%

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

      1. associate-+l- 76.1%

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

      2. sub-neg 76.1%

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

      3. +-commutative 76.1%

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

      4. sub-neg 76.1%

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

      5. distribute-neg-in 76.1%

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

      6. distribute-neg-frac 76.1%

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

      7. metadata-eval 76.1%

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

      8. remove-double-neg 76.1%

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

      9. sub-neg 76.1%

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

      10. metadata-eval 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around inf 76.1%

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

   5. Taylor expanded in x around inf 76.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x}\\ \end{array} \]

Alternative 6: 52.6% 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 87.4%

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

   1. associate-+l- 87.4%

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

   2. sub-neg 87.4%

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

   3. +-commutative 87.4%

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

   4. sub-neg 87.4%

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

   5. distribute-neg-in 87.4%

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

   6. distribute-neg-frac 87.4%

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

   7. metadata-eval 87.4%

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

   8. remove-double-neg 87.4%

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

   9. sub-neg 87.4%

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

   10. metadata-eval 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in x around 0 53.3%

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

5. Final simplification 53.3%

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

Alternative 7: 2.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-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 * -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 * (-x$95$m)), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. associate-+l- 87.4%

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

   2. sub-neg 87.4%

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

   3. +-commutative 87.4%

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

   4. sub-neg 87.4%

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

   5. distribute-neg-in 87.4%

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

   6. distribute-neg-frac 87.4%

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

   7. metadata-eval 87.4%

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

   8. remove-double-neg 87.4%

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

   9. sub-neg 87.4%

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

   10. metadata-eval 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in x around 0 52.4%

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

5. Taylor expanded in x around 0 51.8%

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

   1. neg-mul-1 51.8%

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

   2. associate-*r/ 51.8%

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

   3. metadata-eval 51.8%

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

7. Simplified 51.8%

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

8. Taylor expanded in x around inf 2.9%

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

   1. mul-1-neg 2.9%

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

10. Simplified 2.9%

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

11. Final simplification 2.9%

    \[\leadsto -x \]

Alternative 8: 2.5% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

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
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;
}

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 2.0)
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;
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 * 2.0), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. associate-+l- 87.4%

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

   2. sub-neg 87.4%

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

   3. +-commutative 87.4%

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

   4. sub-neg 87.4%

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

   5. distribute-neg-in 87.4%

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

   6. distribute-neg-frac 87.4%

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

   7. metadata-eval 87.4%

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

   8. remove-double-neg 87.4%

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

   9. sub-neg 87.4%

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

   10. metadata-eval 87.4%

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

3. Simplified 87.4%

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

   1. clear-num 87.4%

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

   2. frac-2neg 87.4%

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

   3. metadata-eval 87.4%

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

   4. frac-add 60.2%

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

   5. *-un-lft-identity 60.2%

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

   6. +-commutative 60.2%

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

   7. distribute-neg-in 60.2%

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

   8. metadata-eval 60.2%

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

   9. sub-neg 60.2%

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

   10. div-inv 60.2%

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

   11. metadata-eval 60.2%

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

   12. div-inv 60.2%

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

   13. metadata-eval 60.2%

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

   14. +-commutative 60.2%

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

   15. distribute-neg-in 60.2%

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

   16. metadata-eval 60.2%

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

   17. sub-neg 60.2%

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

5. Applied egg-rr 60.2%

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)}} \]
6. Step-by-step derivation

   1. associate-*l* 60.2%

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

   2. metadata-eval 60.2%

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

   3. *-commutative 60.2%

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

7. Simplified 60.2%

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

8. Taylor expanded in x around inf 10.7%

   \[\leadsto \frac{1}{1 + x} + \frac{\color{blue}{-0.5 \cdot x}}{\left(1 - x\right) \cdot \left(x \cdot -0.5\right)} \]
9. Step-by-step derivation

   1. *-commutative 10.7%

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

10. Simplified 10.7%

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

11. Taylor expanded in x around 0 3.2%

    \[\leadsto \color{blue}{2} \]

12. Final simplification 3.2%

    \[\leadsto 2 \]

Developer target: 99.5% 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 2023331 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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