2frac (problem 3.3.1)

Percentage Accurate: 77.3% → 99.9%

Time: 6.0s

Alternatives: 6

Speedup: 1.3×

Specification

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]

Initial Program: 77.3% accurate, 1.0× 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]

Alternative 1: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

   1. frac-sub 80.9%

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

   2. *-rgt-identity 80.9%

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

   3. metadata-eval 80.9%

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

   4. div-inv 80.9%

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

   5. associate-/r* 80.9%

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

   6. *-un-lft-identity 80.9%

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

   7. *-rgt-identity 80.9%

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

   8. +-commutative 80.9%

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

   9. div-inv 80.9%

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

   10. metadata-eval 80.9%

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

   11. *-rgt-identity 80.9%

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

   12. +-commutative 80.9%

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

4. Applied egg-rr 80.9%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

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

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(-1.0 / Float64(x * Float64(1.0 + x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

   1. frac-sub 80.9%

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

   2. *-rgt-identity 80.9%

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

   3. metadata-eval 80.9%

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

   4. div-inv 80.9%

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

   5. associate-/r* 80.9%

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

   6. *-un-lft-identity 80.9%

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

   7. *-rgt-identity 80.9%

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

   8. +-commutative 80.9%

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

   9. div-inv 80.9%

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

   10. metadata-eval 80.9%

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

   11. *-rgt-identity 80.9%

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

   12. +-commutative 80.9%

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

4. Applied egg-rr 80.9%

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

   1. *-un-lft-identity 80.9%

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

   2. associate-/l/ 80.9%

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

   3. associate-/r* 80.8%

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

   4. associate--r+ 80.8%

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

   5. div-sub 79.8%

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

   6. div-sub 79.8%

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

   7. *-inverses 79.8%

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

   8. add-exp-log 46.1%

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

   9. sub-neg 46.1%

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

   10. mul-1-neg 46.1%

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

   11. div-inv 46.1%

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

   12. log1p-undefine 46.1%

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

   13. *-inverses 46.1%

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

   14. expm1-undefine 66.1%

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

   15. expm1-log1p-u 99.8%

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

   16. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-/l/ 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.9% accurate, 1.3× 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 79.9%

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

   1. frac-sub 80.9%

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

   2. *-rgt-identity 80.9%

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

   3. metadata-eval 80.9%

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

   4. div-inv 80.9%

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

   5. associate-/r* 80.9%

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

   6. *-un-lft-identity 80.9%

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

   7. *-rgt-identity 80.9%

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

   8. +-commutative 80.9%

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

   9. div-inv 80.9%

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

   10. metadata-eval 80.9%

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

   11. *-rgt-identity 80.9%

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

   12. +-commutative 80.9%

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

4. Applied egg-rr 80.9%

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

   1. associate-/l/ 80.9%

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

   2. associate-/r* 80.8%

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

   3. flip-+ 80.9%

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

   4. associate-/r/ 80.9%

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

   5. associate--r+ 80.9%

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

   6. div-sub 79.9%

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

   7. div-sub 79.9%

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

   8. *-inverses 79.9%

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

   9. add-exp-log 46.1%

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

   10. sub-neg 46.1%

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

   11. mul-1-neg 46.1%

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

   12. div-inv 46.1%

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

   13. log1p-undefine 46.1%

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

   14. *-inverses 46.1%

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

   15. expm1-undefine 60.8%

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

   16. expm1-log1p-u 94.6%

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

   17. metadata-eval 94.6%

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

   18. pow2 94.6%

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

6. Applied egg-rr 94.6%

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

   1. associate-/r/ 99.6%

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

   2. metadata-eval 99.6%

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

   3. unpow2 99.6%

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

   4. flip-+ 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 52.0% accurate, 3.0× 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 79.9%

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

3. Taylor expanded in x around 0 57.2%

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

4. Final simplification 57.2%

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

Alternative 5: 3.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around 0 56.4%

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

   1. div-sub 56.4%

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

   2. sub-neg 56.4%

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

   3. *-commutative 56.4%

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

   4. associate-/l* 56.5%

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

   5. *-inverses 56.5%

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

   6. *-rgt-identity 56.5%

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

   7. neg-mul-1 56.5%

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

   8. unsub-neg 56.5%

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

   9. distribute-neg-frac 56.5%

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

   10. metadata-eval 56.5%

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

5. Simplified 56.5%

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

6. Taylor expanded in x around inf 3.4%

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

   1. neg-mul-1 3.4%

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

8. Simplified 3.4%

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

9. Final simplification 3.4%

   \[\leadsto -x \]
10. Add Preprocessing

Alternative 6: 3.0% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around 0 56.3%

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

4. Taylor expanded in x around inf 2.9%

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

5. Final simplification 2.9%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))