3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.5%

Time: 20.1s

Alternatives: 6

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: 69.3% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({x}^{-3} + {x}^{-5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (+ (pow x -3.0) (pow x -5.0))))

C

double code(double x) {
	return 2.0 * (pow(x, -3.0) + pow(x, -5.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * ((x ** (-3.0d0)) + (x ** (-5.0d0)))
end function

Java

public static double code(double x) {
	return 2.0 * (Math.pow(x, -3.0) + Math.pow(x, -5.0));
}

Python

def code(x):
	return 2.0 * (math.pow(x, -3.0) + math.pow(x, -5.0))

Julia

function code(x)
	return Float64(2.0 * Float64((x ^ -3.0) + (x ^ -5.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * ((x ^ -3.0) + (x ^ -5.0));
end

Wolfram

code[x_] := N[(2.0 * N[(N[Power[x, -3.0], $MachinePrecision] + N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. associate-+l+ 62.9%

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

   3. +-commutative 62.9%

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

3. Simplified 6.0%

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

5. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}} \]
6. Step-by-step derivation

   1. associate-*r/ 98.8%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{5}} \]

   2. metadata-eval 98.8%

      \[\leadsto \frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}} \]

   3. associate-*r/ 98.8%

      \[\leadsto \frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} \]

   4. metadata-eval 98.8%

      \[\leadsto \frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{5}} \]

7. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} + \frac{2}{{x}^{5}} \]

   2. expm1-udef 63.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1\right)} + \frac{2}{{x}^{5}} \]

   3. div-inv 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1\right) + \frac{2}{{x}^{5}} \]

   4. pow-flip 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1\right) + \frac{2}{{x}^{5}} \]

   5. metadata-eval 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1\right) + \frac{2}{{x}^{5}} \]

9. Applied egg-rr 63.1%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1\right)} + \frac{2}{{x}^{5}} \]
10. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} + \frac{2}{{x}^{5}} \]

    2. expm1-log1p 100.0%

       \[\leadsto \color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{5}} \]

11. Simplified 100.0%

    \[\leadsto \color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{5}} \]
12. Step-by-step derivation

    1. expm1-log1p-u 100.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3} + \frac{2}{{x}^{5}}\right)\right)} \]

    2. expm1-udef 62.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-3} + \frac{2}{{x}^{5}}\right)} - 1} \]

    3. fma-def 62.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2, {x}^{-3}, \frac{2}{{x}^{5}}\right)}\right)} - 1 \]

    4. div-inv 62.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-3}, \color{blue}{2 \cdot \frac{1}{{x}^{5}}}\right)\right)} - 1 \]

    5. pow-flip 62.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-3}, 2 \cdot \color{blue}{{x}^{\left(-5\right)}}\right)\right)} - 1 \]

    6. metadata-eval 62.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-3}, 2 \cdot {x}^{\color{blue}{-5}}\right)\right)} - 1 \]

13. Applied egg-rr 62.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-3}, 2 \cdot {x}^{-5}\right)\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-3}, 2 \cdot {x}^{-5}\right)\right)\right)} \]

    2. expm1-log1p 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-3}, 2 \cdot {x}^{-5}\right)} \]

    3. fma-udef 100.0%

       \[\leadsto \color{blue}{2 \cdot {x}^{-3} + 2 \cdot {x}^{-5}} \]

    4. distribute-lft-out 100.0%

       \[\leadsto \color{blue}{2 \cdot \left({x}^{-3} + {x}^{-5}\right)} \]

15. Simplified 100.0%

    \[\leadsto \color{blue}{2 \cdot \left({x}^{-3} + {x}^{-5}\right)} \]

16. Final simplification 100.0%

    \[\leadsto 2 \cdot \left({x}^{-3} + {x}^{-5}\right) \]
17. Add Preprocessing

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. associate-+l+ 62.9%

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

   3. +-commutative 62.9%

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

3. Simplified 6.0%

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

   1. reciprocal-undefine 6.0%

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

   2. frac-sub 5.6%

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

   3. *-rgt-identity 5.6%

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

6. Applied egg-rr 5.6%

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

   1. +-commutative 5.6%

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

   2. reciprocal-undefine 15.7%

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

   3. associate-/r* 63.0%

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

   4. frac-add 63.0%

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

   5. *-un-lft-identity 63.0%

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

   6. +-commutative 63.0%

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

   7. +-commutative 63.0%

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

8. Applied egg-rr 63.0%

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

9. Taylor expanded in x around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 67.8% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. associate-+l+ 62.9%

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

   3. +-commutative 62.9%

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

3. Simplified 6.0%

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

5. Taylor expanded in x around inf 6.0%

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

   1. expm1-log1p-u 6.0%

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

   2. expm1-udef 62.2%

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

   3. sub-div 62.2%

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

   4. metadata-eval 62.2%

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

   5. +-commutative 62.2%

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

7. Applied egg-rr 62.2%

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

   1. expm1-def 6.0%

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

   2. expm1-log1p 6.0%

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

9. Simplified 6.0%

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

    1. div-inv 6.0%

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

    2. fma-def 6.0%

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

    3. reciprocal-define 62.1%

       \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\mathsf{reciprocal}\left(x\right)}, \mathsf{reciprocal}\left(\left(x + 1\right)\right)\right) \]

11. Applied egg-rr 62.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \mathsf{reciprocal}\left(x\right), \mathsf{reciprocal}\left(\left(x + 1\right)\right)\right)} \]
12. Step-by-step derivation

    1. fma-udef 62.1%

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

    2. neg-mul-1 62.1%

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

    3. +-commutative 62.1%

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

    4. sub-neg 62.1%

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

    5. +-commutative 62.1%

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

13. Simplified 62.1%

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

14. Final simplification 62.1%

    \[\leadsto \mathsf{reciprocal}\left(\left(x + 1\right)\right) - \mathsf{reciprocal}\left(x\right) \]
15. Add Preprocessing

Alternative 4: 6.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 4.0 x))

C

double code(double x) {
	return 4.0 / x;
}

Fortran

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

Java

public static double code(double x) {
	return 4.0 / x;
}

Python

def code(x):
	return 4.0 / x

Julia

function code(x)
	return Float64(4.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 4.0 / x;
end

Wolfram

code[x_] := N[(4.0 / x), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. associate-+l+ 62.9%

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

   3. +-commutative 62.9%

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

3. Simplified 6.0%

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

5. Taylor expanded in x around inf 6.1%

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

   1. reciprocal-define 6.0%

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

7. Simplified 6.0%

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

9. Applied egg-rr 6.1%

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

    1. reciprocal-define 6.1%

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

    2. fma-udef 6.1%

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

    3. associate-*r/ 6.1%

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

    4. metadata-eval 6.1%

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

    5. +-commutative 6.1%

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

11. Simplified 6.1%

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

12. Taylor expanded in x around inf 6.1%

    \[\leadsto \color{blue}{\frac{4}{x}} \]

13. Final simplification 6.1%

    \[\leadsto \frac{4}{x} \]
14. Add Preprocessing

Alternative 5: 5.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -\mathsf{reciprocal}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (reciprocal x)))

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. associate-+l+ 62.9%

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

   3. +-commutative 62.9%

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

3. Simplified 6.0%

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

   1. reciprocal-undefine 6.0%

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

   2. frac-sub 5.6%

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

   3. *-rgt-identity 5.6%

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

6. Applied egg-rr 5.6%

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

7. Taylor expanded in x around inf 15.1%

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

8. Taylor expanded in x around 0 4.8%

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

   1. div-inv 4.8%

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

   2. distribute-lft1-in 4.8%

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

   3. metadata-eval 4.8%

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

   4. mul-1-neg 4.8%

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

   5. reciprocal-define 4.8%

      \[\leadsto -\color{blue}{\mathsf{reciprocal}\left(x\right)} \]

10. Applied egg-rr 4.8%

    \[\leadsto \color{blue}{-\mathsf{reciprocal}\left(x\right)} \]

11. Final simplification 4.8%

    \[\leadsto -\mathsf{reciprocal}\left(x\right) \]
12. Add Preprocessing

Alternative 6: 6.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{reciprocal}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (reciprocal x))

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. associate-+l+ 62.9%

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

   3. +-commutative 62.9%

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

3. Simplified 6.0%

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

   1. reciprocal-undefine 6.0%

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

   2. frac-sub 5.6%

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

   3. *-rgt-identity 5.6%

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

6. Applied egg-rr 5.6%

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

7. Taylor expanded in x around inf 15.1%

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

8. Taylor expanded in x around 0 4.8%

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

   1. div-inv 4.8%

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

   2. distribute-lft1-in 4.8%

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

   3. metadata-eval 4.8%

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

   4. div-inv 4.8%

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

   5. add-sqr-sqrt 2.4%

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

   6. sqrt-unprod 45.8%

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

   7. frac-times 48.4%

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

   8. metadata-eval 48.4%

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

   9. metadata-eval 48.4%

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

   10. frac-times 45.8%

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

   11. sqrt-unprod 3.0%

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

   12. add-sqr-sqrt 6.1%

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

   13. *-un-lft-identity 6.1%

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

   14. *-commutative 6.1%

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

   15. reciprocal-define 6.1%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(x\right)} \cdot 1 \]

10. Applied egg-rr 6.1%

    \[\leadsto \color{blue}{\mathsf{reciprocal}\left(x\right) \cdot 1} \]

11. Final simplification 6.1%

    \[\leadsto \mathsf{reciprocal}\left(x\right) \]
12. 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 2024024 
(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))))