3frac (problem 3.3.3)

Percentage Accurate: 69.5% → 99.2%

Time: 10.0s

Alternatives: 8

Speedup: 1.4×

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: 69.5% 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: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. div-inv 72.9%

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

   4. cancel-sign-sub-inv 72.9%

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

   5. metadata-eval 72.9%

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

   6. div-inv 72.9%

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

   7. frac-2neg 72.9%

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

   8. metadata-eval 72.9%

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

   9. sub-neg 72.9%

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

   10. metadata-eval 72.9%

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

   11. distribute-neg-in 72.9%

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

   12. metadata-eval 72.9%

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

   13. +-commutative 72.9%

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

   14. sub-neg 72.9%

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

   15. associate-+l+ 72.9%

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

   16. +-commutative 72.9%

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

   17. frac-add 16.7%

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

4. Applied egg-rr 17.5%

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

5. Taylor expanded in x around 0 99.7%

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{x \cdot \mathsf{fma}\left(x, x, -1\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.7%

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

   2. metadata-eval 99.7%

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

   3. fma-udef 99.7%

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

   4. unpow2 99.7%

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

   5. distribute-rgt-in 99.7%

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

   6. unpow2 99.7%

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

   7. unpow3 99.7%

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

   8. mul-1-neg 99.7%

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

   9. sub-neg 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

    \[\leadsto \frac{2}{{x}^{3} - x} \]
11. Add Preprocessing

Alternative 2: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. div-inv 72.9%

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

   4. cancel-sign-sub-inv 72.9%

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

   5. metadata-eval 72.9%

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

   6. div-inv 72.9%

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

   7. frac-2neg 72.9%

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

   8. metadata-eval 72.9%

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

   9. sub-neg 72.9%

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

   10. metadata-eval 72.9%

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

   11. distribute-neg-in 72.9%

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

   12. metadata-eval 72.9%

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

   13. +-commutative 72.9%

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

   14. sub-neg 72.9%

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

   15. associate-+l+ 72.9%

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

   16. +-commutative 72.9%

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

   17. frac-add 16.7%

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

4. Applied egg-rr 17.5%

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

5. Taylor expanded in x around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 67.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in x around inf 71.6%

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

4. Final simplification 71.6%

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

Alternative 4: 67.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in x around inf 71.6%

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

5. Applied egg-rr 71.6%

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

   1. +-commutative 71.6%

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

7. Simplified 71.6%

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

8. Final simplification 71.6%

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

Alternative 5: 51.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in x around inf 71.6%

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

5. Applied egg-rr 71.7%

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

   1. *-rgt-identity 71.7%

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

   2. associate-+r+ 59.0%

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

   3. *-commutative 59.0%

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

   4. mul-1-neg 59.0%

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

   5. sub-neg 59.0%

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

   6. +-inverses 59.0%

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

   7. metadata-eval 59.0%

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

   8. associate-*l* 59.0%

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

   9. *-commutative 59.0%

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

   10. associate-/r* 58.3%

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

   11. *-commutative 58.3%

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

   12. neg-mul-1 58.3%

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

   13. distribute-neg-in 58.3%

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

   14. metadata-eval 58.3%

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

   15. unsub-neg 58.3%

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

7. Simplified 58.3%

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

8. Final simplification 58.3%

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

Alternative 6: 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 72.9%

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

3. Taylor expanded in x around 0 5.3%

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

4. Final simplification 5.3%

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

Alternative 7: 5.0% 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 72.9%

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

3. Taylor expanded in x around inf 71.6%

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

4. Taylor expanded in x around 0 5.3%

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

5. Final simplification 5.3%

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

Alternative 8: 6.3% 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 72.9%

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

4. Applied egg-rr 71.8%

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

   1. expm1-def 6.3%

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

   2. expm1-log1p 6.3%

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

   3. +-commutative 6.3%

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

   4. +-commutative 6.3%

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

6. Simplified 6.3%

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

7. Taylor expanded in x around 0 6.3%

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

8. Taylor expanded in x around 0 6.3%

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

9. Final simplification 6.3%

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

Developer target: 99.2% 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 2024011 
(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))))