3frac (problem 3.3.3)

Percentage Accurate: 69.8% → 99.8%

Time: 12.2s

Alternatives: 6

Speedup: 1.2×

Specification

\[\left|x\right| > 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% 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: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left(-0.5 \cdot \left(1 - x\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 x) (* (+ 1.0 x) (* -0.5 (- 1.0 x)))))

C

double code(double x) {
	return (1.0 / x) / ((1.0 + x) * (-0.5 * (1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) / ((1.0d0 + x) * ((-0.5d0) * (1.0d0 - x)))
end function

Java

public static double code(double x) {
	return (1.0 / x) / ((1.0 + x) * (-0.5 * (1.0 - x)));
}

Python

def code(x):
	return (1.0 / x) / ((1.0 + x) * (-0.5 * (1.0 - x)))

Julia

function code(x)
	return Float64(Float64(1.0 / x) / Float64(Float64(1.0 + x) * Float64(-0.5 * Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) / ((1.0 + x) * (-0.5 * (1.0 - x)));
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * N[(-0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

   1. associate-+l- 68.7%

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

   2. sub-neg 68.7%

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

   3. +-commutative 68.7%

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

   4. neg-sub0 68.7%

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

   5. associate-+l- 68.7%

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

   6. neg-sub0 68.7%

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

   7. distribute-neg-frac 68.7%

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

   8. metadata-eval 68.7%

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

   9. sub-neg 68.7%

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

   10. metadata-eval 68.7%

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

3. Simplified 68.7%

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

   1. clear-num 68.7%

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

   2. frac-2neg 68.7%

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

   3. metadata-eval 68.7%

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

   4. frac-add 22.7%

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

   5. *-un-lft-identity 22.7%

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

   6. +-commutative 22.7%

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

   7. distribute-neg-in 22.7%

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

   8. metadata-eval 22.7%

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

   9. sub-neg 22.7%

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

   10. div-inv 22.7%

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

   11. metadata-eval 22.7%

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

   12. div-inv 22.7%

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

   13. metadata-eval 22.7%

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

   14. +-commutative 22.7%

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

   15. distribute-neg-in 22.7%

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

   16. metadata-eval 22.7%

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

   17. sub-neg 22.7%

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

6. Applied egg-rr 22.7%

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

   1. associate-*l* 22.7%

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

   2. associate-+l- 22.7%

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

   3. *-commutative 22.7%

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

   4. associate-*r* 22.7%

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

   5. neg-mul-1 22.7%

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

   6. cancel-sign-sub 22.7%

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

   7. +-commutative 22.7%

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

   8. *-commutative 22.7%

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

   9. distribute-lft1-in 22.7%

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

   10. metadata-eval 22.7%

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

   11. metadata-eval 22.7%

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

   12. *-commutative 22.7%

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

   13. metadata-eval 22.7%

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

8. Simplified 22.7%

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

   1. associate-/r* 68.7%

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

   2. frac-add 68.7%

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

   3. *-un-lft-identity 68.7%

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

   4. +-commutative 68.7%

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

   5. +-commutative 68.7%

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

10. Applied egg-rr 68.7%

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

11. Taylor expanded in x around 0 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left(-0.5 \cdot \left(1 - x\right)\right)} \]
13. Add Preprocessing

Alternative 2: 68.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (-2.0 / x) + ((1.0 / x) + (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 / (x + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. sub-neg 68.7%

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

   2. distribute-neg-frac 68.7%

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

   3. metadata-eval 68.7%

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

   4. metadata-eval 68.7%

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

   5. metadata-eval 68.7%

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

   6. associate-/r* 68.7%

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

   7. metadata-eval 68.7%

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

   8. neg-mul-1 68.7%

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

   9. +-commutative 68.7%

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

   10. associate-+l+ 68.7%

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

   11. +-commutative 68.7%

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

   12. neg-mul-1 68.7%

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

   13. metadata-eval 68.7%

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

   14. associate-/r* 68.7%

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

   15. metadata-eval 68.7%

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

   16. metadata-eval 68.7%

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

   17. +-commutative 68.7%

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

   18. +-commutative 68.7%

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

   19. sub-neg 68.7%

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

   20. metadata-eval 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around inf 66.4%

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

6. Final simplification 66.4%

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

Alternative 3: 68.1% 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 68.7%

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

   1. sub-neg 68.7%

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

   2. distribute-neg-frac 68.7%

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

   3. metadata-eval 68.7%

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

   4. metadata-eval 68.7%

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

   5. metadata-eval 68.7%

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

   6. associate-/r* 68.7%

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

   7. metadata-eval 68.7%

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

   8. neg-mul-1 68.7%

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

   9. +-commutative 68.7%

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

   10. associate-+l+ 68.7%

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

   11. +-commutative 68.7%

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

   12. neg-mul-1 68.7%

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

   13. metadata-eval 68.7%

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

   14. associate-/r* 68.7%

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

   15. metadata-eval 68.7%

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

   16. metadata-eval 68.7%

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

   17. +-commutative 68.7%

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

   18. +-commutative 68.7%

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

   19. sub-neg 68.7%

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

   20. metadata-eval 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around inf 66.2%

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

6. Final simplification 66.2%

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

Alternative 4: 5.1% 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 68.7%

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

   1. sub-neg 68.7%

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

   2. distribute-neg-frac 68.7%

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

   3. metadata-eval 68.7%

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

   4. metadata-eval 68.7%

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

   5. metadata-eval 68.7%

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

   6. associate-/r* 68.7%

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

   7. metadata-eval 68.7%

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

   8. neg-mul-1 68.7%

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

   9. +-commutative 68.7%

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

   10. associate-+l+ 68.7%

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

   11. +-commutative 68.7%

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

   12. neg-mul-1 68.7%

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

   13. metadata-eval 68.7%

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

   14. associate-/r* 68.7%

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

   15. metadata-eval 68.7%

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

   16. metadata-eval 68.7%

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

   17. +-commutative 68.7%

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

   18. +-commutative 68.7%

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

   19. sub-neg 68.7%

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

   20. metadata-eval 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around 0 4.8%

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

6. Final simplification 4.8%

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

Alternative 5: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. sub-neg 68.7%

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

   2. distribute-neg-frac 68.7%

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

   3. metadata-eval 68.7%

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

   4. metadata-eval 68.7%

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

   5. metadata-eval 68.7%

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

   6. associate-/r* 68.7%

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

   7. metadata-eval 68.7%

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

   8. neg-mul-1 68.7%

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

   9. +-commutative 68.7%

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

   10. associate-+l+ 68.7%

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

   11. +-commutative 68.7%

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

   12. neg-mul-1 68.7%

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

   13. metadata-eval 68.7%

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

   14. associate-/r* 68.7%

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

   15. metadata-eval 68.7%

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

   16. metadata-eval 68.7%

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

   17. +-commutative 68.7%

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

   18. +-commutative 68.7%

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

   19. sub-neg 68.7%

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

   20. metadata-eval 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around inf 3.4%

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

7. Taylor expanded in x around 0 4.8%

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

8. Final simplification 4.8%

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

Alternative 6: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 68.7%

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

   1. sub-neg 68.7%

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

   2. distribute-neg-frac 68.7%

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

   3. metadata-eval 68.7%

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

   4. metadata-eval 68.7%

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

   5. metadata-eval 68.7%

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

   6. associate-/r* 68.7%

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

   7. metadata-eval 68.7%

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

   8. neg-mul-1 68.7%

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

   9. +-commutative 68.7%

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

   10. associate-+l+ 68.7%

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

   11. +-commutative 68.7%

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

   12. neg-mul-1 68.7%

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

   13. metadata-eval 68.7%

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

   14. associate-/r* 68.7%

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

   15. metadata-eval 68.7%

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

   16. metadata-eval 68.7%

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

   17. +-commutative 68.7%

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

   18. +-commutative 68.7%

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

   19. sub-neg 68.7%

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

   20. metadata-eval 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around inf 3.4%

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

7. Final simplification 3.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 99.3% 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 2024022 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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