2frac (problem 3.3.1)

Percentage Accurate: 76.9% → 99.9%

Time: 36.9s

Alternatives: 5

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: 76.9% 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

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

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 76.4%

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

   1. sub-neg 76.4%

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

   2. add-exp-log 51.6%

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

   3. log-rec 51.6%

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

   4. +-commutative 51.6%

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

   5. log1p-define 51.6%

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

   6. distribute-neg-frac 51.6%

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

   7. metadata-eval 51.6%

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

4. Applied egg-rr 51.6%

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

   1. *-rgt-identity 51.6%

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

   2. fma-undefine 51.6%

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

   3. *-inverses 51.6%

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

   4. metadata-eval 51.6%

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

   5. distribute-neg-frac 51.6%

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

   6. fma-neg 51.6%

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

   7. *-inverses 51.6%

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

   8. *-rgt-identity 51.6%

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

   9. exp-neg 51.6%

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

   10. log1p-undefine 51.6%

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

   11. rem-exp-log 76.4%

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

   12. *-inverses 76.4%

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

   13. associate-/l/ 54.2%

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

   14. associate-/r* 76.4%

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

   15. div-sub 76.4%

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

   16. *-inverses 76.4%

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

   17. div-sub 77.0%

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

   18. associate-/l/ 77.0%

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

   19. +-commutative 77.0%

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

   20. associate--r+ 99.3%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.75))) (/ (/ -1.0 x) x) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = ((-1.0d0) / x) / x
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.75):
		tmp = (-1.0 / x) / x
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.75))
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.75)))
		tmp = (-1.0 / x) / x;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 52.9%

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

   3. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

      2. associate-/r* 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1} - \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. sub-neg 76.4%

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

   2. add-exp-log 51.6%

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

   3. log-rec 51.6%

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

   4. +-commutative 51.6%

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

   5. log1p-define 51.6%

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

   6. distribute-neg-frac 51.6%

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

   7. metadata-eval 51.6%

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

4. Applied egg-rr 51.6%

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

   1. *-rgt-identity 51.6%

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

   2. fma-undefine 51.6%

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

   3. *-inverses 51.6%

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

   4. metadata-eval 51.6%

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

   5. distribute-neg-frac 51.6%

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

   6. fma-neg 51.6%

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

   7. *-inverses 51.6%

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

   8. *-rgt-identity 51.6%

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

   9. exp-neg 51.6%

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

   10. log1p-undefine 51.6%

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

   11. rem-exp-log 76.4%

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

   12. *-inverses 76.4%

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

   13. associate-/l/ 54.2%

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

   14. associate-/r* 76.4%

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

   15. div-sub 76.4%

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

   16. *-inverses 76.4%

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

   17. div-sub 77.0%

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

   18. associate-/l/ 77.0%

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

   19. +-commutative 77.0%

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

   20. associate--r+ 99.3%

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

6. Simplified 99.9%

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

   1. div-inv 99.9%

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

   2. associate-/l* 99.9%

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

   3. add-sqr-sqrt 42.0%

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

   4. sqrt-unprod 51.2%

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

   5. frac-times 51.1%

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

   6. metadata-eval 51.1%

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

   7. metadata-eval 51.1%

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

   8. frac-times 51.2%

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

   9. sqrt-unprod 17.4%

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

   10. add-sqr-sqrt 25.4%

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

   11. div-inv 25.4%

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

   12. metadata-eval 25.4%

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

   13. associate-/r* 25.4%

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

   14. associate-*l/ 25.4%

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

   15. *-un-lft-identity 25.4%

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

   16. add-exp-log 8.2%

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

   17. rec-exp 8.2%

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

   18. +-commutative 8.2%

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

   19. log1p-define 8.2%

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

   20. add-sqr-sqrt 0.2%

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

   21. sqrt-unprod 33.0%

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

   22. mul-1-neg 33.0%

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

   23. mul-1-neg 33.0%

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

   24. sqr-neg 33.0%

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

8. Applied egg-rr 68.2%

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

   1. associate-*r/ 68.2%

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

   2. associate-*l/ 68.2%

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

   3. exp-neg 68.2%

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

   4. log1p-undefine 68.2%

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

   5. +-commutative 68.2%

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

   6. rem-exp-log 99.8%

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

   7. associate-*r/ 99.9%

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

   8. *-rgt-identity 99.9%

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

   9. associate-/r* 99.3%

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

10. Simplified 99.3%

    \[\leadsto \color{blue}{\frac{-1}{x \cdot \left(x + 1\right)}} \]
11. 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 76.4%

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

3. Taylor expanded in x around 0 51.8%

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

Alternative 5: 3.8% 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 76.4%

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

3. Taylor expanded in x around inf 50.4%

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

   1. unpow2 50.4%

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

   2. associate-/r* 50.9%

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

5. Applied egg-rr 50.9%

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

   1. div-inv 50.8%

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

   2. frac-2neg 50.8%

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

   3. metadata-eval 50.8%

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

   4. frac-times 50.4%

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

   5. metadata-eval 50.4%

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

7. Applied egg-rr 50.4%

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

9. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{\frac{x}{\left(0 + \left(x + 0 \cdot x\right)\right) \cdot -1}} \]
10. Step-by-step derivation

    1. +-lft-identity 3.6%

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

    2. mul0-lft 3.6%

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

    3. +-rgt-identity 3.6%

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

    4. associate-/r* 3.6%

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

    5. *-inverses 3.6%

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

    6. metadata-eval 3.6%

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

11. Simplified 3.6%

    \[\leadsto \color{blue}{-1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))