3frac (problem 3.3.3)

Percentage Accurate: 69.7% → 99.5%

Time: 8.6s

Alternatives: 12

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: 69.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 99.0%

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

   1. associate-*r/ 99.0%

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

   2. metadata-eval 99.0%

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

7. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. +-commutative 99.0%

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

   3. div-inv 99.0%

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

   4. fma-define 99.0%

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

   5. pow-flip 99.0%

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

   6. metadata-eval 99.0%

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

   7. pow-flip 99.5%

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

   8. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

Alternative 2: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x \cdot x} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ 2.0 (* x x)) (+ 2.0 (/ 2.0 (pow x 4.0)))) (pow x 3.0)))

C

double code(double x) {
	return ((2.0 / (x * x)) + (2.0 + (2.0 / pow(x, 4.0)))) / pow(x, 3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 / (x * x)) + (2.0d0 + (2.0d0 / (x ** 4.0d0)))) / (x ** 3.0d0)
end function

Java

public static double code(double x) {
	return ((2.0 / (x * x)) + (2.0 + (2.0 / Math.pow(x, 4.0)))) / Math.pow(x, 3.0);
}

Python

def code(x):
	return ((2.0 / (x * x)) + (2.0 + (2.0 / math.pow(x, 4.0)))) / math.pow(x, 3.0)

Julia

function code(x)
	return Float64(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 + Float64(2.0 / (x ^ 4.0)))) / (x ^ 3.0))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 / (x * x)) + (2.0 + (2.0 / (x ^ 4.0)))) / (x ^ 3.0);
end

Wolfram

code[x_] := N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(2.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-+r+ 99.1%

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

   2. +-commutative 99.1%

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

   3. associate-+l+ 99.1%

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

   4. associate-*r/ 99.1%

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

   5. metadata-eval 99.1%

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

7. Simplified 99.1%

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

   1. unpow2 99.1%

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

9. Applied egg-rr 99.1%

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

Alternative 3: 99.0% 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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 99.0%

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

   1. associate-*r/ 99.0%

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

   2. metadata-eval 99.0%

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

7. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. +-commutative 99.0%

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

   3. div-inv 99.0%

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

   4. fma-define 99.0%

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

   5. pow-flip 99.0%

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

   6. metadata-eval 99.0%

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

   7. pow-flip 99.5%

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

   8. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

10. Taylor expanded in x around inf 99.1%

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

Alternative 4: 69.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 + ((-1.0d0) / x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 66.1%

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

6. Final simplification 66.1%

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

Alternative 5: 69.7% accurate, 1.0× speedup

Math

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

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(1.0 / Float64(x + 1.0)) + Float64(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[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

5. Final simplification 66.1%

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

Alternative 6: 69.7% 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 66.1%

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

3. Final simplification 66.1%

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

Alternative 7: 68.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.9%

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

   1. *-un-lft-identity 64.9%

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

   2. associate-+r+ 64.9%

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

7. Applied egg-rr 64.9%

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

   1. *-lft-identity 64.9%

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

   2. +-commutative 64.9%

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

   3. metadata-eval 64.9%

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

   4. associate-*r/ 64.9%

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

   5. distribute-rgt1-in 64.9%

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

   6. metadata-eval 64.9%

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

   7. associate-*r/ 64.9%

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

   8. metadata-eval 64.9%

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

9. Simplified 64.9%

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

Alternative 8: 68.1% accurate, 2.1× speedup

Math

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

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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.9%

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

6. Taylor expanded in x around inf 64.7%

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

Alternative 9: 52.1% accurate, 2.1× speedup

Math

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

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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.9%

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

6. Taylor expanded in x around inf 64.9%

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

   1. *-un-lft-identity 64.9%

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

   2. add-sqr-sqrt 24.4%

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

   3. sqrt-prod 12.2%

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

   4. frac-times 11.8%

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

   5. metadata-eval 11.8%

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

   6. metadata-eval 11.8%

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

   7. frac-times 12.2%

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

   8. sqrt-unprod 2.0%

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

   9. add-sqr-sqrt 5.0%

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

   10. div-sub 5.0%

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

   11. inv-pow 5.0%

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

   12. pow1 5.0%

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

   13. pow-div 5.0%

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

   14. metadata-eval 5.0%

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

   15. add-sqr-sqrt 3.0%

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

   16. sqrt-prod 9.7%

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

   17. frac-times 8.9%

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

   18. metadata-eval 8.9%

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

   19. metadata-eval 8.9%

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

   20. frac-times 9.7%

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

   21. sqrt-unprod 13.4%

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

   22. add-sqr-sqrt 64.9%

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

8. Applied egg-rr 64.9%

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

   1. *-lft-identity 64.9%

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

   2. +-commutative 64.9%

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

   3. sub-neg 64.9%

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

   4. exp-to-pow 39.3%

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

   5. *-commutative 39.3%

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

   6. metadata-eval 39.3%

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

   7. distribute-lft-neg-in 39.3%

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

   8. exp-neg 39.3%

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

   9. *-commutative 39.3%

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

   10. exp-to-pow 64.9%

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

   11. unpow2 64.9%

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

   12. associate-/r* 64.9%

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

   13. metadata-eval 64.9%

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

   14. associate-*l/ 64.9%

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

   15. associate-*r/ 64.9%

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

   16. neg-mul-1 64.9%

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

   17. distribute-rgt-in 64.9%

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

   18. *-rgt-identity 64.9%

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

   19. distribute-lft-out 64.9%

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

   20. +-commutative 64.9%

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

10. Simplified 64.9%

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

11. Taylor expanded in x around 0 53.4%

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

12. Final simplification 53.4%

    \[\leadsto \frac{1}{x} \cdot \frac{1}{x} \]
13. Add Preprocessing

Alternative 10: 6.3% 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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.9%

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

6. Taylor expanded in x around 0 5.0%

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

   1. div-inv 5.0%

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

   2. add-sqr-sqrt 3.0%

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

   3. sqrt-prod 51.4%

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

   4. frac-times 53.6%

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

   5. metadata-eval 53.6%

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

   6. metadata-eval 53.6%

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

   7. frac-times 51.4%

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

   8. sqrt-unprod 2.6%

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

   9. add-sqr-sqrt 6.3%

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

8. Applied egg-rr 6.3%

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

9. Taylor expanded in x around 0 6.3%

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

Alternative 11: 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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around inf 64.9%

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

6. Taylor expanded in x around 0 5.0%

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

Alternative 12: 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 66.1%

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

3. Simplified 66.1%

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

5. Taylor expanded in x around 0 5.0%

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

Developer Target 1: 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 2024139 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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