2frac (problem 3.3.1)

Percentage Accurate: 77.8% → 99.4%

Time: 7.2s

Alternatives: 7

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: 77.8% 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: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. frac-sub N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \left(x + 1\right) \cdot 1\right), \color{blue}{\left(\left(x + 1\right) \cdot x\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \left(x + 1\right) \cdot \frac{1}{1}\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \frac{x + 1}{1}\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x + \left(\mathsf{neg}\left(\frac{x + 1}{1}\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} \cdot x\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{x + 1}{1}\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) \cdot x\right)\right) \]

   7. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(x + 1\right) \cdot \frac{1}{1}\right)\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(x + 1\right) \cdot 1\right)\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} \cdot x\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) + -1\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   14. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-1 - x\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   15. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\left(\left(x + 1\right) \cdot 1\right) \cdot x\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\left(\left(x + 1\right) \cdot \frac{1}{1}\right) \cdot x\right)\right) \]

   18. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\frac{x + 1}{1} \cdot x\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(x \cdot \color{blue}{\frac{x + 1}{1}}\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x + 1}{1}\right)}\right)\right) \]

   21. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{\frac{1}{1}}\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot 1\right)\right)\right) \]

   23. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

   24. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(1 + \color{blue}{x}\right)\right)\right) \]

   25. +-lowering-+.f64 79.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

4. Applied egg-rr 79.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{-1}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.8%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x + \color{blue}{1}\right)\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot x + \color{blue}{x \cdot 1}\right)\right) \]

   3. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot x + x\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{x}\right)\right) \]

   5. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), x\right)\right) \]

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-1.0d0) / x) / x
    else if (x <= 1.0d0) then
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    else
        tmp = (-1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-1.0 / x) / x
	elif x <= 1.0:
		tmp = (1.0 - x) + (-1.0 / x)
	else:
		tmp = -1.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	else
		tmp = Float64(-1.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-1.0 / x) / x;
	elseif (x <= 1.0)
		tmp = (1.0 - x) + (-1.0 / x);
	else
		tmp = -1.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 58.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 95.6%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 95.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. neg-mul-1 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right), \color{blue}{x}\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{1}{x}\right), x\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{x}\right), x\right) \]

      7. /-lowering-/.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), x\right) \]

   7. Applied egg-rr 96.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}, \mathsf{/.f64}\left(1, x\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 - x\right), \mathsf{/.f64}\left(\color{blue}{1}, x\right)\right) \]

      3. --lowering--.f64 99.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\color{blue}{1}, x\right)\right) \]

   5. Simplified 99.6%

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

   if 1 < x

   1. Initial program 49.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 99.8%

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

4. Final simplification 98.8%

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

Alternative 3: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ -1.0 x) x)
   (if (<= x 0.75) (+ 1.0 (/ -1.0 x)) (/ -1.0 (* x x)))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-1.0d0) / x) / x
    else if (x <= 0.75d0) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = (-1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-1.0 / x) / x
	elif x <= 0.75:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = -1.0 / (x * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.75], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 58.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 95.6%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 95.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. neg-mul-1 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right), \color{blue}{x}\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{1}{x}\right), x\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{x}\right), x\right) \]

      7. /-lowering-/.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), x\right) \]

   7. Applied egg-rr 96.1%

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

   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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.6%

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

   if 0.75 < x

   1. Initial program 49.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* x x))))
   (if (<= x -1.0) t_0 (if (<= x 0.75) (+ 1.0 (/ -1.0 x)) t_0))))

C

double code(double x) {
	double t_0 = -1.0 / (x * x);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.75) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / (x * x)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 0.75d0) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -1.0 / (x * x);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.75) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -1.0 / (x * x)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 0.75:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-1.0 / Float64(x * x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.75)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -1.0 / (x * x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.75)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.75], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 54.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 97.4%

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

   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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 78.7%

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

   1. frac-sub N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \left(x + 1\right) \cdot 1\right), \color{blue}{\left(\left(x + 1\right) \cdot x\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \left(x + 1\right) \cdot \frac{1}{1}\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \frac{x + 1}{1}\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x + \left(\mathsf{neg}\left(\frac{x + 1}{1}\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} \cdot x\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{x + 1}{1}\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) \cdot x\right)\right) \]

   7. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(x + 1\right) \cdot \frac{1}{1}\right)\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(x + 1\right) \cdot 1\right)\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} \cdot x\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(x\right)\right) + -1\right)\right), \left(\left(x + 1\right) \cdot x\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   14. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(-1 - x\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   15. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\left(x + \color{blue}{1}\right) \cdot x\right)\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\left(\left(x + 1\right) \cdot 1\right) \cdot x\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\left(\left(x + 1\right) \cdot \frac{1}{1}\right) \cdot x\right)\right) \]

   18. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(\frac{x + 1}{1} \cdot x\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \left(x \cdot \color{blue}{\frac{x + 1}{1}}\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x + 1}{1}\right)}\right)\right) \]

   21. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{\frac{1}{1}}\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot 1\right)\right)\right) \]

   23. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

   24. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \left(1 + \color{blue}{x}\right)\right)\right) \]

   25. +-lowering-+.f64 79.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

4. Applied egg-rr 79.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{-1}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 6: 51.8% 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 78.7%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 55.0%

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{x}\right) \]

5. Simplified 55.0%

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

Alternative 7: 2.9% 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 78.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
4. Step-by-step derivation

5. Simplified 54.1%

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

6. Taylor expanded in x around inf

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

8. Simplified 3.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))