3frac (problem 3.3.3)

Percentage Accurate: 84.5% → 99.6%

Time: 7.5s

Alternatives: 5

Speedup: 1.4×

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: 84.5% 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: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. associate-+l- 84.9%

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

   2. sub-neg 84.9%

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

   3. neg-mul-1 84.9%

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

   4. metadata-eval 84.9%

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

   5. cancel-sign-sub-inv 84.9%

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

   6. +-commutative 84.9%

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

   7. *-lft-identity 84.9%

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

   8. sub-neg 84.9%

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

   9. metadata-eval 84.9%

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

3. Simplified 84.9%

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

   1. frac-2neg 84.9%

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

   2. metadata-eval 84.9%

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

   3. frac-sub 59.8%

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

   4. +-commutative 59.8%

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

   5. distribute-neg-in 59.8%

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

   6. metadata-eval 59.8%

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

   7. sub-neg 59.8%

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

   8. *-commutative 59.8%

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

   9. neg-mul-1 59.8%

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

   10. +-commutative 59.8%

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

   11. distribute-neg-in 59.8%

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

   12. metadata-eval 59.8%

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

   13. sub-neg 59.8%

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

5. Applied egg-rr 59.8%

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

   1. frac-sub 60.2%

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

   2. *-un-lft-identity 60.2%

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

7. Applied egg-rr 60.2%

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

8. Taylor expanded in x around 0 99.5%

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

   1. expm1-log1p-u 75.7%

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

   2. expm1-udef 27.7%

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

10. Applied egg-rr 27.7%

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

    1. expm1-def 75.7%

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

    2. expm1-log1p 99.5%

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

    3. *-commutative 99.5%

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

    4. associate-*l* 99.6%

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

    5. +-commutative 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{\left(x \cdot x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.85) (not (<= x 1.0)))
   (/ -2.0 (* (* x x) (- -1.0 x)))
   (- (* -2.0 x) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.85) || !(x <= 1.0)) {
		tmp = -2.0 / ((x * x) * (-1.0 - x));
	} else {
		tmp = (-2.0 * x) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.85d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-2.0d0) / ((x * x) * ((-1.0d0) - x))
    else
        tmp = ((-2.0d0) * x) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.85) || !(x <= 1.0)) {
		tmp = -2.0 / ((x * x) * (-1.0 - x));
	} else {
		tmp = (-2.0 * x) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.85) or not (x <= 1.0):
		tmp = -2.0 / ((x * x) * (-1.0 - x))
	else:
		tmp = (-2.0 * x) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.85) || !(x <= 1.0))
		tmp = Float64(-2.0 / Float64(Float64(x * x) * Float64(-1.0 - x)));
	else
		tmp = Float64(Float64(-2.0 * x) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.85) || ~((x <= 1.0)))
		tmp = -2.0 / ((x * x) * (-1.0 - x));
	else
		tmp = (-2.0 * x) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.85], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-2.0 / N[(N[(x * x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.849999999999999978 or 1 < x

   1. Initial program 69.3%

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

      1. associate-+l- 69.3%

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

      2. sub-neg 69.3%

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

      3. neg-mul-1 69.3%

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

      4. metadata-eval 69.3%

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

      5. cancel-sign-sub-inv 69.3%

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

      6. +-commutative 69.3%

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

      7. *-lft-identity 69.3%

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

      8. sub-neg 69.3%

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

      9. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

      1. frac-2neg 69.3%

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

      2. metadata-eval 69.3%

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

      3. frac-sub 18.3%

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

      4. +-commutative 18.3%

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

      5. distribute-neg-in 18.3%

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

      6. metadata-eval 18.3%

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

      7. sub-neg 18.3%

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

      8. *-commutative 18.3%

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

      9. neg-mul-1 18.3%

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

      10. +-commutative 18.3%

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

      11. distribute-neg-in 18.3%

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

      12. metadata-eval 18.3%

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

      13. sub-neg 18.3%

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

   5. Applied egg-rr 18.3%

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

      1. frac-sub 19.1%

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

      2. *-un-lft-identity 19.1%

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

   7. Applied egg-rr 19.1%

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

   8. Taylor expanded in x around 0 99.1%

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

   9. Taylor expanded in x around inf 95.9%

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

       1. mul-1-neg 95.9%

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

       2. unpow2 95.9%

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

       3. distribute-rgt-neg-in 95.9%

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

   11. Simplified 95.9%

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

   if -0.849999999999999978 < x < 1

   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. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. 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. Taylor expanded in x around 0 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{\left(x \cdot x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \end{array} \]

Alternative 3: 75.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.15e+77)))
   (/ -1.0 (* x x))
   (- 1.0 (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.15e+77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.15d+77))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.15e+77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.15e+77):
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.15e+77))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.15e+77)))
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.15e+77]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.14999999999999997e77 < x

   1. Initial program 81.8%

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

      1. associate-+l- 81.8%

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

      2. sub-neg 81.8%

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

      3. neg-mul-1 81.8%

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

      4. metadata-eval 81.8%

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

      5. cancel-sign-sub-inv 81.8%

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

      6. +-commutative 81.8%

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

      7. *-lft-identity 81.8%

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

      8. sub-neg 81.8%

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

      9. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in x around inf 81.8%

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

   5. Taylor expanded in x around inf 66.3%

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

      1. unpow2 66.3%

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

   7. Simplified 66.3%

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

   if -1 < x < 1.14999999999999997e77

   1. Initial program 86.9%

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

      1. associate-+l- 86.9%

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

      2. sub-neg 86.9%

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

      3. neg-mul-1 86.9%

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

      4. metadata-eval 86.9%

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

      5. cancel-sign-sub-inv 86.9%

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

      6. +-commutative 86.9%

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

      7. *-lft-identity 86.9%

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

      8. sub-neg 86.9%

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

      9. metadata-eval 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around 0 84.4%

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

   5. Taylor expanded in x around 0 84.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{x}\\ \end{array} \]

Alternative 4: 83.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. associate-+l- 84.9%

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

   2. sub-neg 84.9%

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

   3. neg-mul-1 84.9%

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

   4. metadata-eval 84.9%

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

   5. cancel-sign-sub-inv 84.9%

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

   6. +-commutative 84.9%

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

   7. *-lft-identity 84.9%

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

   8. sub-neg 84.9%

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

   9. metadata-eval 84.9%

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

3. Simplified 84.9%

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

4. Taylor expanded in x around 0 51.7%

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

5. Taylor expanded in x around 0 83.0%

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

6. Final simplification 83.0%

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

Alternative 5: 51.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. associate-+l- 84.9%

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

   2. sub-neg 84.9%

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

   3. neg-mul-1 84.9%

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

   4. metadata-eval 84.9%

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

   5. cancel-sign-sub-inv 84.9%

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

   6. +-commutative 84.9%

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

   7. *-lft-identity 84.9%

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

   8. sub-neg 84.9%

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

   9. metadata-eval 84.9%

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

3. Simplified 84.9%

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

4. Taylor expanded in x around 0 52.6%

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

5. Final simplification 52.6%

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

Developer target: 99.6% 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 2023223 
(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))))