3frac (problem 3.3.3)

Percentage Accurate: 68.7% → 99.9%

Time: 12.4s

Alternatives: 10

Speedup: 1.0×

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: 68.7% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x\_m + 1} - \frac{2}{x\_m}\right) + \frac{1}{x\_m + -1} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {x\_m}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \left(x\_m + 1\right) \cdot \left(x\_m + -1\right), x\_m \cdot \left(x\_m + \left(\left(x\_m + 1\right) + -1\right)\right)\right)}{\left(x\_m + -1\right) \cdot \left(x\_m + {x\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (+ (- (/ 1.0 (+ x_m 1.0)) (/ 2.0 x_m)) (/ 1.0 (+ x_m -1.0))) 5e-25)
    (* 2.0 (pow x_m -3.0))
    (/
     (fma
      -2.0
      (* (+ x_m 1.0) (+ x_m -1.0))
      (* x_m (+ x_m (+ (+ x_m 1.0) -1.0))))
     (* (+ x_m -1.0) (+ x_m (pow x_m 2.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-25) {
		tmp = 2.0 * pow(x_m, -3.0);
	} else {
		tmp = fma(-2.0, ((x_m + 1.0) * (x_m + -1.0)), (x_m * (x_m + ((x_m + 1.0) + -1.0)))) / ((x_m + -1.0) * (x_m + pow(x_m, 2.0)));
	}
	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 (Float64(Float64(Float64(1.0 / Float64(x_m + 1.0)) - Float64(2.0 / x_m)) + Float64(1.0 / Float64(x_m + -1.0))) <= 5e-25)
		tmp = Float64(2.0 * (x_m ^ -3.0));
	else
		tmp = Float64(fma(-2.0, Float64(Float64(x_m + 1.0) * Float64(x_m + -1.0)), Float64(x_m * Float64(x_m + Float64(Float64(x_m + 1.0) + -1.0)))) / Float64(Float64(x_m + -1.0) * Float64(x_m + (x_m ^ 2.0))));
	end
	return Float64(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[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-25], N[(2.0 * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(x$95$m + 1.0), $MachinePrecision] * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m + N[(N[(x$95$m + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m + -1.0), $MachinePrecision] * N[(x$95$m + N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 4.99999999999999962e-25

   1. Initial program 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 98.0%

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

      1. clear-num 98.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{2}}} \]

      2. associate-/r/ 98.0%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

      3. pow-flip 98.6%

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

      4. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 4.99999999999999962e-25 < (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 88.7%

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

   3. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. +-commutative 88.7%

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

      3. associate-+l+ 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. *-lft-identity 93.0%

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

      2. +-commutative 93.0%

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

   8. Simplified 93.0%

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

      1. frac-add 93.0%

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

      2. frac-add 98.4%

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

      3. *-un-lft-identity 98.4%

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

   10. Applied egg-rr 98.4%

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

       1. fma-define 98.4%

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

       2. *-rgt-identity 98.4%

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

       3. +-commutative 98.4%

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

       4. associate-+l+ 98.4%

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

       5. *-commutative 98.4%

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

       6. associate-*l* 100.0%

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

       7. distribute-lft1-in 100.0%

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

       8. unpow2 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \left(x + 1\right) \cdot \left(x + -1\right), x \cdot \left(x + \left(\left(x + 1\right) + -1\right)\right)\right)}{\left(x + -1\right) \cdot \left(x + {x}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\left(2 + 2 \cdot {x\_m}^{-2}\right) \cdot \left({x\_m}^{-4} + 1\right)\right) \cdot {x\_m}^{-3}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (* (+ 2.0 (* 2.0 (pow x_m -2.0))) (+ (pow x_m -4.0) 1.0))
   (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 + (2.0 * pow(x_m, -2.0))) * (pow(x_m, -4.0) + 1.0)) * 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 + (2.0d0 * (x_m ** (-2.0d0)))) * ((x_m ** (-4.0d0)) + 1.0d0)) * (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 + (2.0 * Math.pow(x_m, -2.0))) * (Math.pow(x_m, -4.0) + 1.0)) * 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 + (2.0 * math.pow(x_m, -2.0))) * (math.pow(x_m, -4.0) + 1.0)) * 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(Float64(Float64(2.0 + Float64(2.0 * (x_m ^ -2.0))) * Float64((x_m ^ -4.0) + 1.0)) * (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 + (2.0 * (x_m ^ -2.0))) * ((x_m ^ -4.0) + 1.0)) * (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[(N[(N[(2.0 + N[(2.0 * N[Power[x$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[x$95$m, -4.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in x around -inf 98.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}{{x}^{3}}} \]

7. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{2 + \frac{2}{{x}^{2}}}{{x}^{4}} + \left(2 + \frac{2}{{x}^{2}}\right)}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \color{blue}{\left(\frac{2 + \frac{2}{{x}^{2}}}{{x}^{4}} + \left(2 + \frac{2}{{x}^{2}}\right)\right) \cdot \frac{1}{{x}^{3}}} \]

   2. div-inv 98.7%

      \[\leadsto \left(\color{blue}{\left(2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{4}}} + \left(2 + \frac{2}{{x}^{2}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   3. fma-define 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 + \frac{2}{{x}^{2}}, \frac{1}{{x}^{4}}, 2 + \frac{2}{{x}^{2}}\right)} \cdot \frac{1}{{x}^{3}} \]

   4. +-commutative 98.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{{x}^{2}} + 2}, \frac{1}{{x}^{4}}, 2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}} \]

   5. div-inv 98.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + 2, \frac{1}{{x}^{4}}, 2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}} \]

   6. fma-define 98.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2\right)}, \frac{1}{{x}^{4}}, 2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}} \]

   7. pow-flip 98.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2\right), \frac{1}{{x}^{4}}, 2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}} \]

   8. metadata-eval 98.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2\right), \frac{1}{{x}^{4}}, 2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}} \]

   9. pow-flip 98.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), \color{blue}{{x}^{\left(-4\right)}}, 2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}} \]

   10. metadata-eval 98.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{\color{blue}{-4}}, 2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}} \]

   11. +-commutative 98.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \color{blue}{\frac{2}{{x}^{2}} + 2}\right) \cdot \frac{1}{{x}^{3}} \]

   12. div-inv 98.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \color{blue}{2 \cdot \frac{1}{{x}^{2}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

   13. fma-define 98.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2\right)}\right) \cdot \frac{1}{{x}^{3}} \]

   14. pow-flip 98.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   15. metadata-eval 98.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   16. pow-flip 99.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \mathsf{fma}\left(2, {x}^{-2}, 2\right)\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   17. metadata-eval 99.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \mathsf{fma}\left(2, {x}^{-2}, 2\right)\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right), {x}^{-4}, \mathsf{fma}\left(2, {x}^{-2}, 2\right)\right) \cdot {x}^{-3}} \]
10. Step-by-step derivation

    1. fma-undefine 99.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-4} + \mathsf{fma}\left(2, {x}^{-2}, 2\right)\right)} \cdot {x}^{-3} \]

    2. *-rgt-identity 99.3%

       \[\leadsto \left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-4} + \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot 1}\right) \cdot {x}^{-3} \]

    3. distribute-lft-out 99.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot \left({x}^{-4} + 1\right)\right)} \cdot {x}^{-3} \]

11. Simplified 99.3%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot \left({x}^{-4} + 1\right)\right) \cdot {x}^{-3}} \]
12. Step-by-step derivation

    1. fma-undefine 99.3%

       \[\leadsto \left(\color{blue}{\left(2 \cdot {x}^{-2} + 2\right)} \cdot \left({x}^{-4} + 1\right)\right) \cdot {x}^{-3} \]

13. Applied egg-rr 99.3%

    \[\leadsto \left(\color{blue}{\left(2 \cdot {x}^{-2} + 2\right)} \cdot \left({x}^{-4} + 1\right)\right) \cdot {x}^{-3} \]

14. Final simplification 99.3%

    \[\leadsto \left(\left(2 + 2 \cdot {x}^{-2}\right) \cdot \left({x}^{-4} + 1\right)\right) \cdot {x}^{-3} \]
15. Add Preprocessing

Alternative 3: 99.3% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in x around inf 98.3%

   \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-*r/ 98.3%

      \[\leadsto \frac{2 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}}{{x}^{3}} \]

   2. metadata-eval 98.3%

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

7. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

      \[\leadsto \frac{\color{blue}{1 \cdot \left(2 + \frac{2}{{x}^{2}}\right)}}{{x}^{3}} \]

   2. cube-mult 98.2%

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

   3. unpow2 98.2%

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

   4. times-frac 98.7%

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

   5. +-commutative 98.7%

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

   6. div-inv 98.7%

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

   7. fma-define 98.7%

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

   8. pow-flip 98.7%

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

   9. metadata-eval 98.7%

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

9. Applied egg-rr 98.7%

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

Alternative 4: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m + 1\right) \cdot \left(x\_m + -1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x\_m + 1} - \frac{2}{x\_m}\right) + \frac{1}{x\_m + -1} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {x\_m}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(x\_m + 1\right) + \left(x\_m + -1\right)\right) + -2 \cdot t\_0}{x\_m \cdot t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (+ x_m 1.0) (+ x_m -1.0))))
   (*
    x_s
    (if (<= (+ (- (/ 1.0 (+ x_m 1.0)) (/ 2.0 x_m)) (/ 1.0 (+ x_m -1.0))) 5e-25)
      (* 2.0 (pow x_m -3.0))
      (/ (+ (* x_m (+ (+ x_m 1.0) (+ x_m -1.0))) (* -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 = (x_m + 1.0) * (x_m + -1.0);
	double tmp;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-25) {
		tmp = 2.0 * pow(x_m, -3.0);
	} else {
		tmp = ((x_m * ((x_m + 1.0) + (x_m + -1.0))) + (-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 = (x_m + 1.0d0) * (x_m + (-1.0d0))
    if ((((1.0d0 / (x_m + 1.0d0)) - (2.0d0 / x_m)) + (1.0d0 / (x_m + (-1.0d0)))) <= 5d-25) then
        tmp = 2.0d0 * (x_m ** (-3.0d0))
    else
        tmp = ((x_m * ((x_m + 1.0d0) + (x_m + (-1.0d0)))) + ((-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 = (x_m + 1.0) * (x_m + -1.0);
	double tmp;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-25) {
		tmp = 2.0 * Math.pow(x_m, -3.0);
	} else {
		tmp = ((x_m * ((x_m + 1.0) + (x_m + -1.0))) + (-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 = (x_m + 1.0) * (x_m + -1.0)
	tmp = 0
	if (((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-25:
		tmp = 2.0 * math.pow(x_m, -3.0)
	else:
		tmp = ((x_m * ((x_m + 1.0) + (x_m + -1.0))) + (-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(x_m + 1.0) * Float64(x_m + -1.0))
	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-25)
		tmp = Float64(2.0 * (x_m ^ -3.0));
	else
		tmp = Float64(Float64(Float64(x_m * Float64(Float64(x_m + 1.0) + Float64(x_m + -1.0))) + 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 = (x_m + 1.0) * (x_m + -1.0);
	tmp = 0.0;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 5e-25)
		tmp = 2.0 * (x_m ^ -3.0);
	else
		tmp = ((x_m * ((x_m + 1.0) + (x_m + -1.0))) + (-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[(x$95$m + 1.0), $MachinePrecision] * N[(x$95$m + -1.0), $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-25], N[(2.0 * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(N[(x$95$m + 1.0), $MachinePrecision] + N[(x$95$m + -1.0), $MachinePrecision]), $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 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 4.99999999999999962e-25

   1. Initial program 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 98.0%

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

      1. clear-num 98.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{2}}} \]

      2. associate-/r/ 98.0%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

      3. pow-flip 98.6%

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

      4. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 4.99999999999999962e-25 < (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 88.7%

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

   3. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. +-commutative 88.7%

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

      3. associate-+l+ 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. *-lft-identity 93.0%

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

      2. +-commutative 93.0%

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

   8. Simplified 93.0%

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

      1. +-commutative 93.0%

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

      2. frac-add 93.0%

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

      3. frac-add 98.4%

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

      4. *-un-lft-identity 98.4%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x + 1\right) + \left(x + -1\right)\right) + -2 \cdot \left(\left(x + 1\right) \cdot \left(x + -1\right)\right)}{x \cdot \left(\left(x + 1\right) \cdot \left(x + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (/ 1.0 (+ x_m 1.0)) (+ (/ -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 / (x_m + 1.0)) + ((-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 / (x_m + 1.0d0)) + (((-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 / (x_m + 1.0)) + ((-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 / (x_m + 1.0)) + ((-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(1.0 / Float64(x_m + 1.0)) + Float64(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 / (x_m + 1.0)) + ((-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[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / x$95$m), $MachinePrecision] + N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Simplified 73.1%

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

5. Final simplification 73.1%

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

Alternative 6: 68.7% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (- (/ 1.0 (+ x_m 1.0)) (/ 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 / (x_m + 1.0)) - (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 / (x_m + 1.0d0)) - (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 / (x_m + 1.0)) - (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 / (x_m + 1.0)) - (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(x_m + 1.0)) - 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 / (x_m + 1.0)) - (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[(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]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Final simplification 73.1%

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

Alternative 7: 67.6% accurate, 1.2× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((1.0 / (x_m + 1.0)) + (((1.0 / 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(1.0 / Float64(x_m + 1.0)) + Float64(Float64(Float64(1.0 / x_m) + -1.0) / x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((1.0 / (x_m + 1.0)) + (((1.0 / 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[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + -1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in x around inf 71.4%

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

6. Final simplification 71.4%

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

Alternative 8: 67.1% 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 #s(literal 1 binary64) 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 73.1%

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

3. Simplified 73.1%

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

   1. *-un-lft-identity 73.1%

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

   2. +-commutative 73.1%

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

   3. associate-+l+ 73.1%

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

6. Applied egg-rr 73.1%

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

   1. *-lft-identity 73.1%

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

   2. +-commutative 73.1%

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

8. Simplified 73.1%

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

9. Taylor expanded in x around inf 70.9%

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

10. Final simplification 70.9%

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

Alternative 9: 5.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in x around inf 71.3%

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

6. Taylor expanded in x around 0 5.2%

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

Alternative 10: 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 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ -2.0 x_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in x around 0 5.2%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024090 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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