3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.5%

Time: 8.7s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. associate-+l- 72.7%

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

   2. sub-neg 72.7%

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

   3. +-commutative 72.7%

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

   4. neg-sub0 72.7%

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

   5. associate-+l- 72.7%

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

   6. neg-sub0 72.7%

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

   7. distribute-neg-frac 72.7%

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

   8. metadata-eval 72.7%

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

   9. sub-neg 72.7%

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

   10. metadata-eval 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

   3. associate-*r/ 99.2%

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

   4. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 71.5%

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

   3. div-inv 71.5%

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

   4. fma-def 71.5%

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

   5. pow-flip 71.5%

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

   6. metadata-eval 71.5%

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

   7. div-inv 71.5%

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

   8. pow-flip 71.5%

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

   9. metadata-eval 71.5%

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

9. Applied egg-rr 71.5%

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

    1. expm1-def 99.7%

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

    2. expm1-log1p 99.7%

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

    3. fma-udef 99.7%

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

    4. +-commutative 99.7%

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

    5. distribute-lft-out 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. associate-+l- 72.7%

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

   2. sub-neg 72.7%

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

   3. +-commutative 72.7%

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

   4. neg-sub0 72.7%

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

   5. associate-+l- 72.7%

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

   6. neg-sub0 72.7%

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

   7. distribute-neg-frac 72.7%

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

   8. metadata-eval 72.7%

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

   9. sub-neg 72.7%

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

   10. metadata-eval 72.7%

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

3. Simplified 72.7%

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

   1. flip-+ 21.1%

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

   2. sub-neg 21.1%

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

   3. metadata-eval 21.1%

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

   4. distribute-neg-in 21.1%

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

   5. +-commutative 21.1%

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

   6. associate-/r/ 18.1%

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

   7. fma-def 7.5%

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

   8. metadata-eval 7.5%

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

   9. pow2 7.5%

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

   10. +-commutative 7.5%

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

   11. distribute-neg-in 7.5%

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

   12. metadata-eval 7.5%

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

   13. sub-neg 7.5%

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

6. Applied egg-rr 7.5%

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

   1. fma-udef 18.1%

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

   2. associate-*l/ 18.8%

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

   3. *-lft-identity 18.8%

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

8. Simplified 18.8%

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

   1. +-commutative 18.8%

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

   2. frac-add 72.7%

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

   3. clear-num 70.4%

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

   4. metadata-eval 70.4%

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

   5. unpow2 70.4%

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

   6. flip-+ 19.0%

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

   7. frac-add 19.5%

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

   8. fma-def 19.5%

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

   9. +-commutative 19.5%

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

   10. +-commutative 19.5%

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

10. Applied egg-rr 19.5%

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

11. Taylor expanded in x around 0 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 68.0% 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 72.7%

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

   1. associate-+l- 72.7%

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

   2. sub-neg 72.7%

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

   3. +-commutative 72.7%

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

   4. neg-sub0 72.7%

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

   5. associate-+l- 72.7%

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

   6. neg-sub0 72.7%

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

   7. distribute-neg-frac 72.7%

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

   8. metadata-eval 72.7%

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

   9. sub-neg 72.7%

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

   10. metadata-eval 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around inf 71.4%

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

   1. expm1-log1p-u 71.4%

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

   2. expm1-udef 71.3%

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

7. Applied egg-rr 71.3%

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

   1. expm1-def 71.4%

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

   2. expm1-log1p 71.4%

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

   3. associate-+r+ 71.4%

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

   4. +-commutative 71.4%

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

   5. associate-+l+ 71.4%

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

9. Simplified 71.4%

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

    1. expm1-log1p-u 71.4%

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

    2. expm1-udef 71.3%

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

    3. associate-+r+ 71.3%

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

    4. +-commutative 71.3%

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

    5. div-inv 71.3%

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

    6. *-un-lft-identity 71.3%

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

    7. distribute-rgt-out 71.3%

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

    8. metadata-eval 71.3%

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

11. Applied egg-rr 71.3%

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

    1. expm1-def 71.4%

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

    2. expm1-log1p 71.4%

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

    3. +-commutative 71.4%

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

    4. associate-*l/ 71.4%

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

    5. metadata-eval 71.4%

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

13. Simplified 71.4%

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

14. Final simplification 71.4%

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

Alternative 4: 67.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. associate-+l- 72.7%

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

   2. sub-neg 72.7%

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

   3. +-commutative 72.7%

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

   4. neg-sub0 72.7%

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

   5. associate-+l- 72.7%

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

   6. neg-sub0 72.7%

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

   7. distribute-neg-frac 72.7%

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

   8. metadata-eval 72.7%

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

   9. sub-neg 72.7%

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

   10. metadata-eval 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around 0 3.3%

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

6. Taylor expanded in x around 0 71.1%

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

   1. +-commutative 71.1%

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

   2. distribute-neg-in 71.1%

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

   3. associate-*r/ 71.1%

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

   4. metadata-eval 71.1%

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

   5. distribute-neg-frac 71.1%

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

   6. metadata-eval 71.1%

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

   7. metadata-eval 71.1%

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

8. Simplified 71.1%

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

9. Final simplification 71.1%

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

Alternative 5: 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 72.7%

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

   1. associate-+l- 72.7%

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

   2. sub-neg 72.7%

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

   3. +-commutative 72.7%

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

   4. neg-sub0 72.7%

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

   5. associate-+l- 72.7%

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

   6. neg-sub0 72.7%

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

   7. distribute-neg-frac 72.7%

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

   8. metadata-eval 72.7%

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

   9. sub-neg 72.7%

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

   10. metadata-eval 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around 0 5.4%

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

6. Final simplification 5.4%

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

Alternative 6: 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 72.7%

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

   1. associate-+l- 72.7%

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

   2. sub-neg 72.7%

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

   3. +-commutative 72.7%

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

   4. neg-sub0 72.7%

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

   5. associate-+l- 72.7%

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

   6. neg-sub0 72.7%

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

   7. distribute-neg-frac 72.7%

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

   8. metadata-eval 72.7%

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

   9. sub-neg 72.7%

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

   10. metadata-eval 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around inf 71.4%

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

6. Taylor expanded in x around 0 5.4%

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

7. Final simplification 5.4%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(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))))