2frac (problem 3.3.1)

Percentage Accurate: 77.3% → 99.9%

Time: 8.6s

Alternatives: 7

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× 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 / Float64(x + 1.0)) / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. frac-sub N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. *-lft-identity N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. neg-lowering-neg.f64 N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. /-lowering-/.f64 N/A

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

4. Applied egg-rr 81.4%

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

6. Applied egg-rr 83.0%

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate--r+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-lowering-+.f64 99.9

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

8. Applied egg-rr 99.9%

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

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2000000000:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ x 1.0)) (/ -1.0 x))))
   (if (<= t_0 -2000000000.0)
     (+ 1.0 (/ -1.0 x))
     (if (<= t_0 0.0) (/ -1.0 (* x x)) (+ (- 1.0 x) (/ -1.0 x))))))

C

double code(double x) {
	double t_0 = (1.0 / (x + 1.0)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2000000000.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	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 + 1.0d0)) + ((-1.0d0) / x)
    if (t_0 <= (-2000000000.0d0)) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (x + 1.0)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2000000000.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (x + 1.0)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -2000000000.0:
		tmp = 1.0 + (-1.0 / x)
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = (1.0 - x) + (-1.0 / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2000000000.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (x + 1.0)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2000000000.0)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = (1.0 - x) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2e9

   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 100.0%

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

   if -2e9 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 60.4%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{{x}^{2} + 0}} \]

      3. unpow2 N/A

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

      4. accelerator-lowering-fma.f64 95.9

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

   5. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{fma}\left(x, x, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 95.9

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

   7. Applied egg-rr 95.9%

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

   if 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. --lowering--.f64 99.5

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 - x\right)} - \frac{1}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{x} \leq -2000000000:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{x + 1} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2000000000:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ x 1.0)) (/ -1.0 x))))
   (if (<= t_0 -2000000000.0)
     (+ 1.0 (/ -1.0 x))
     (if (<= t_0 0.0) (/ -1.0 (* x x)) (/ (+ -1.0 x) x)))))

C

double code(double x) {
	double t_0 = (1.0 / (x + 1.0)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2000000000.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = (-1.0 + x) / x;
	}
	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 + 1.0d0)) + ((-1.0d0) / x)
    if (t_0 <= (-2000000000.0d0)) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = ((-1.0d0) + x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (x + 1.0)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2000000000.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = (-1.0 + x) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (x + 1.0)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -2000000000.0:
		tmp = 1.0 + (-1.0 / x)
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = (-1.0 + x) / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2000000000.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(Float64(-1.0 + x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (x + 1.0)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2000000000.0)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = (-1.0 + x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2e9

   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 100.0%

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

   if -2e9 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 60.4%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{{x}^{2} + 0}} \]

      3. unpow2 N/A

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

      4. accelerator-lowering-fma.f64 95.9

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

   5. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{fma}\left(x, x, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 95.9

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

   7. Applied egg-rr 95.9%

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

   if 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. Step-by-step derivation

      1. frac-sub N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-lft-identity N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. frac-times N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. distribute-lft1-in N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate--r+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-lowering-+.f64 100.0

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. +-lowering-+.f64 98.2

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

   11. Simplified 98.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{x} \leq -2000000000:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{x + 1} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 81.4%

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

   1. frac-sub N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. *-lft-identity N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. neg-lowering-neg.f64 N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. /-lowering-/.f64 N/A

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

4. Applied egg-rr 81.4%

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

6. Applied egg-rr 83.0%

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

   1. associate--r+ N/A

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

   2. +-inverses N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval 99.2

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

8. Applied egg-rr 99.2%

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

Alternative 5: 51.3% 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 81.4%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 54.5

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

5. Simplified 54.5%

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

Alternative 6: 3.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ 0 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 0.0 x))

C

double code(double x) {
	return 0.0 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0 - x
end function

Java

public static double code(double x) {
	return 0.0 - x;
}

Python

def code(x):
	return 0.0 - x

Julia

function code(x)
	return Float64(0.0 - x)
end

MATLAB

function tmp = code(x)
	tmp = 0.0 - x;
end

Wolfram

code[x_] := N[(0.0 - x), $MachinePrecision]

Derivation

1. Initial program 81.4%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
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. --lowering--.f64 54.1

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

5. Simplified 54.1%

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

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. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x} \]

   3. --lowering--.f64 3.1

      \[\leadsto \color{blue}{0 - x} \]

8. Simplified 3.1%

   \[\leadsto \color{blue}{0 - x} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 3.1

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

10. Applied egg-rr 3.1%

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

11. Final simplification 3.1%

    \[\leadsto 0 - x \]
12. Add Preprocessing

Alternative 7: 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 81.4%

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

3. Taylor expanded in x around 0

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

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

   6. *-rgt-identity N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. neg-sub0 N/A

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

   11. associate-+l- N/A

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

   12. neg-sub0 N/A

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

   13. mul-1-neg N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

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

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

   19. mul-1-neg N/A

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

   20. neg-sub0 N/A

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

   21. associate-+l- N/A

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

   22. neg-sub0 N/A

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

   23. +-commutative N/A

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

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

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

   25. distribute-neg-frac N/A

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

   26. metadata-eval N/A

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

   27. /-lowering-/.f64 53.8

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

5. Simplified 53.8%

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

6. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-neg-out N/A

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

   6. lft-mult-inverse N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt1-in N/A

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

   11. lft-mult-inverse N/A

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

   12. associate-+r+ N/A

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

8. Simplified 2.5%

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

9. Taylor expanded in x around 0

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

11. Simplified 3.1%

    \[\leadsto \color{blue}{1} \]
12. 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 2024194 
(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)))