3frac (problem 3.3.3)

Percentage Accurate: 68.2% → 98.9%

Time: 10.5s

Alternatives: 9

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: 68.2% 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: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (pow x -3.0)))

C

double code(double x) {
	return 2.0 * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return 2.0 * Math.pow(x, -3.0);
}

Python

def code(x):
	return 2.0 * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(2.0 * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x ^ -3.0);
end

Wolfram

code[x_] := N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 99.1%

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

   1. div-inv 99.1%

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

   2. pow-flip 99.6%

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

   3. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]
8. Add Preprocessing

Alternative 2: 71.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{2}{x} + \frac{1}{1 - x}\right) + \frac{3 + \frac{1}{x}}{x}}{\left(x + 1\right) \cdot \frac{x \cdot \left(x + 1\right) - 2}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+ (+ (/ 2.0 x) (/ 1.0 (- 1.0 x))) (/ (+ 3.0 (/ 1.0 x)) x))
  (* (+ x 1.0) (/ (- (* x (+ x 1.0)) 2.0) x))))

C

double code(double x) {
	return (((2.0 / x) + (1.0 / (1.0 - x))) + ((3.0 + (1.0 / x)) / x)) / ((x + 1.0) * (((x * (x + 1.0)) - 2.0) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((2.0d0 / x) + (1.0d0 / (1.0d0 - x))) + ((3.0d0 + (1.0d0 / x)) / x)) / ((x + 1.0d0) * (((x * (x + 1.0d0)) - 2.0d0) / x))
end function

Java

public static double code(double x) {
	return (((2.0 / x) + (1.0 / (1.0 - x))) + ((3.0 + (1.0 / x)) / x)) / ((x + 1.0) * (((x * (x + 1.0)) - 2.0) / x));
}

Python

def code(x):
	return (((2.0 / x) + (1.0 / (1.0 - x))) + ((3.0 + (1.0 / x)) / x)) / ((x + 1.0) * (((x * (x + 1.0)) - 2.0) / x))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(2.0 / x) + Float64(1.0 / Float64(1.0 - x))) + Float64(Float64(3.0 + Float64(1.0 / x)) / x)) / Float64(Float64(x + 1.0) * Float64(Float64(Float64(x * Float64(x + 1.0)) - 2.0) / x)))
end

MATLAB

function tmp = code(x)
	tmp = (((2.0 / x) + (1.0 / (1.0 - x))) + ((3.0 + (1.0 / x)) / x)) / ((x + 1.0) * (((x * (x + 1.0)) - 2.0) / x));
end

Wolfram

code[x_] := N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.3%

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

3. Simplified 65.3%

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

   1. flip-+ 17.9%

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

   2. frac-add 17.1%

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

   3. *-un-lft-identity 17.1%

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

   4. frac-times 13.7%

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

   5. metadata-eval 13.7%

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

   6. pow2 13.7%

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

   7. inv-pow 13.7%

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

   8. inv-pow 13.7%

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

   9. pow-prod-up 14.5%

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

   10. metadata-eval 14.5%

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

6. Applied egg-rr 14.5%

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

7. Taylor expanded in x around inf 48.1%

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

8. Taylor expanded in x around 0 65.0%

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

   1. sub-neg 65.0%

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

   2. add-sqr-sqrt 30.4%

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

   3. sqrt-unprod 67.3%

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

   4. frac-times 67.3%

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

   5. metadata-eval 67.3%

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

   6. unpow2 67.3%

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

   7. sqrt-div 67.3%

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

   8. metadata-eval 67.3%

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

   9. sqrt-pow1 69.8%

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

   10. metadata-eval 69.8%

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

   11. pow1 69.8%

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

10. Applied egg-rr 69.8%

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

    1. distribute-neg-frac2 69.8%

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

    2. +-commutative 69.8%

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

    3. distribute-neg-in 69.8%

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

    4. metadata-eval 69.8%

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

    5. sub-neg 69.8%

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

12. Simplified 69.8%

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

13. Final simplification 69.8%

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

Alternative 3: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Simplified 65.3%

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

   1. *-un-lft-identity 65.3%

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

   2. +-commutative 65.3%

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

   3. associate-+l+ 65.4%

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

6. Applied egg-rr 65.4%

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

   1. *-lft-identity 65.4%

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

   2. +-commutative 65.4%

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

8. Simplified 65.4%

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

Alternative 4: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Final simplification 65.3%

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

Alternative 5: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} + \frac{\frac{1}{x} + -1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 65.0%

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

6. Final simplification 65.0%

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

Alternative 6: 66.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Simplified 65.3%

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

   1. *-un-lft-identity 65.3%

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

   2. +-commutative 65.3%

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

   3. associate-+l+ 65.4%

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

6. Applied egg-rr 65.4%

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

   1. *-lft-identity 65.4%

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

   2. +-commutative 65.4%

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

8. Simplified 65.4%

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

9. Taylor expanded in x around inf 64.8%

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

10. Final simplification 64.8%

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

Alternative 7: 5.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -0.5 x))

C

double code(double x) {
	return -0.5 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.5d0) / x
end function

Java

public static double code(double x) {
	return -0.5 / x;
}

Python

def code(x):
	return -0.5 / x

Julia

function code(x)
	return Float64(-0.5 / x)
end

MATLAB

function tmp = code(x)
	tmp = -0.5 / x;
end

Wolfram

code[x_] := N[(-0.5 / x), $MachinePrecision]

Derivation

1. Initial program 65.3%

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

3. Simplified 65.3%

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

   1. flip-+ 17.9%

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

   2. frac-add 17.1%

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

   3. *-un-lft-identity 17.1%

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

   4. frac-times 13.7%

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

   5. metadata-eval 13.7%

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

   6. pow2 13.7%

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

   7. inv-pow 13.7%

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

   8. inv-pow 13.7%

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

   9. pow-prod-up 14.5%

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

   10. metadata-eval 14.5%

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

6. Applied egg-rr 14.5%

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

7. Taylor expanded in x around inf 48.1%

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

8. Taylor expanded in x around 0 4.8%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
9. Add Preprocessing

Alternative 8: 5.0% 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 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around 0 4.7%

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

Alternative 9: 3.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

double code(double x) {
	return -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

function tmp = code(x)
	tmp = -0.5;
end

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 65.3%

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

3. Simplified 65.3%

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

   1. flip-+ 17.9%

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

   2. frac-add 17.1%

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

   3. *-un-lft-identity 17.1%

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

   4. frac-times 13.7%

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

   5. metadata-eval 13.7%

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

   6. pow2 13.7%

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

   7. inv-pow 13.7%

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

   8. inv-pow 13.7%

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

   9. pow-prod-up 14.5%

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

   10. metadata-eval 14.5%

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

6. Applied egg-rr 14.5%

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

7. Taylor expanded in x around inf 48.0%

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

8. Taylor expanded in x around 0 3.4%

   \[\leadsto \color{blue}{-0.5} \]
9. 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 2024102 
(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))))