2frac (problem 3.3.1)

Percentage Accurate: 77.7% → 99.9%

Time: 8.9s

Alternatives: 10

Speedup: 1.6×

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: 77.7% 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, 1.1× 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 79.1%

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

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. *-lft-identity N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied egg-rr 79.1%

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

   1. lift-fma.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. un-div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 80.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 80.2

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

6. Applied egg-rr 80.2%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate--r+ N/A

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

   10. +-inverses N/A

       \[\leadsto \frac{\frac{\color{blue}{0} - 1}{x + 1}}{x} \]

   11. metadata-eval N/A

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

   12. lower-/.f64 99.9

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

8. Applied egg-rr 99.9%

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

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. metadata-eval N/A

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

   14. unsub-neg N/A

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

   15. lower--.f64 99.9

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

10. Applied egg-rr 99.9%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := \left(1 - x\right) + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))) (t_1 (+ (- 1.0 x) (/ -1.0 x))))
   (if (<= t_0 -500.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = (1.0d0 - x) + ((-1.0d0) / x)
    if (t_0 <= (-500.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = (1.0 - x) + (-1.0 / x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	t_1 = Float64(Float64(1.0 - x) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = (1.0 - x) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.9

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

   5. Simplified 98.9%

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

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 53.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := 1 + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))) (t_1 (+ 1.0 (/ -1.0 x))))
   (if (<= t_0 -500.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = 1.0 + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = 1.0d0 + ((-1.0d0) / x)
    if (t_0 <= (-500.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = 1.0 + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = 1.0 + (-1.0 / x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	t_1 = Float64(1.0 + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = 1.0 + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 53.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.1× 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 79.1%

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

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. *-lft-identity N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied egg-rr 79.1%

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

   1. lift-fma.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. un-div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 80.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 80.2

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

6. Applied egg-rr 80.2%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate--r+ N/A

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

   10. +-inverses N/A

       \[\leadsto \frac{\frac{\color{blue}{0} - 1}{x + 1}}{x} \]

   11. metadata-eval N/A

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

   12. lower-/.f64 99.9

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 5: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(x, x, x\right)} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return Float64(-1.0 / fma(x, x, x))
end

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. *-lft-identity N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied egg-rr 79.1%

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

   1. lift-fma.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. un-div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 80.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 80.2

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

6. Applied egg-rr 80.2%

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

   1. associate--r+ N/A

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

   2. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0} - 1}{\mathsf{fma}\left(x, x, x\right)} \]

   3. metadata-eval 99.7

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

8. Applied egg-rr 99.7%

   \[\leadsto \frac{\color{blue}{-1}}{\mathsf{fma}\left(x, x, x\right)} \]
9. Add Preprocessing

Alternative 6: 51.9% accurate, 2.4× 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 79.1%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 56.5

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

5. Simplified 56.5%

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

Alternative 7: 3.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

4. Applied egg-rr 23.0%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 24.6

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

7. Simplified 24.6%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 3.2

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

10. Simplified 3.2%

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

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{-2 \cdot x} \]
12. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f64 3.3

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

13. Simplified 3.3%

    \[\leadsto \color{blue}{x \cdot -2} \]
14. Add Preprocessing

Alternative 8: 3.2% accurate, 9.7× 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 79.1%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 55.6

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

5. Simplified 55.6%

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

6. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 3.3

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

8. Simplified 3.3%

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

Alternative 9: 3.0% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 79.1%

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

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

4. Applied egg-rr 23.0%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 24.6

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

7. Simplified 24.6%

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

8. Taylor expanded in x around 0

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

10. Simplified 3.2%

    \[\leadsto \color{blue}{2} \]
11. Add Preprocessing

Alternative 10: 3.0% accurate, 29.0× 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 79.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 55.5%

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

6. Taylor expanded in x around inf

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

8. Simplified 3.2%

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

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

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

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

MATLAB

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

Wolfram

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

Developer Target 2: 99.4% accurate, 1.5× 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]

Reproduce

herbie shell --seed 2024208 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ -1 x) (+ x 1)))

  :alt
  (! :herbie-platform default (/ 1 (* x (- -1 x))))

  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))