2frac (problem 3.3.1)

Percentage Accurate: 77.9% → 99.9%

Time: 6.0s

Alternatives: 7

Speedup: 1.0×

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: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} - \frac{1}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-{x}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (+ 1.0 x)) (/ 1.0 x))))
   (if (<= t_0 -5e-22)
     (/ (- x (+ 1.0 x)) (* x (+ 1.0 x)))
     (if (<= t_0 0.0) (- (pow x -2.0)) t_0))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) - (1.0 / x);
	double tmp;
	if (t_0 <= -5e-22) {
		tmp = (x - (1.0 + x)) / (x * (1.0 + x));
	} else if (t_0 <= 0.0) {
		tmp = -pow(x, -2.0);
	} 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 / (1.0d0 + x)) - (1.0d0 / x)
    if (t_0 <= (-5d-22)) then
        tmp = (x - (1.0d0 + x)) / (x * (1.0d0 + x))
    else if (t_0 <= 0.0d0) then
        tmp = -(x ** (-2.0d0))
    else
        tmp = t_0
    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 tmp;
	if (t_0 <= -5e-22) {
		tmp = (x - (1.0 + x)) / (x * (1.0 + x));
	} else if (t_0 <= 0.0) {
		tmp = -Math.pow(x, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) - (1.0 / x)
	tmp = 0
	if t_0 <= -5e-22:
		tmp = (x - (1.0 + x)) / (x * (1.0 + x))
	elif t_0 <= 0.0:
		tmp = -math.pow(x, -2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(1.0 / x))
	tmp = 0.0
	if (t_0 <= -5e-22)
		tmp = Float64(Float64(x - Float64(1.0 + x)) / Float64(x * Float64(1.0 + x)));
	elseif (t_0 <= 0.0)
		tmp = Float64(-(x ^ -2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) - (1.0 / x);
	tmp = 0.0;
	if (t_0 <= -5e-22)
		tmp = (x - (1.0 + x)) / (x * (1.0 + x));
	elseif (t_0 <= 0.0)
		tmp = -(x ^ -2.0);
	else
		tmp = t_0;
	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]}, If[LessEqual[t$95$0, -5e-22], N[(N[(x - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-N[Power[x, -2.0], $MachinePrecision]), t$95$0]]]

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)) < -4.99999999999999954e-22

   1. Initial program 96.9%

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

      1. clear-num 96.9%

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

      2. frac-sub 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. div-inv 99.9%

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

      5. metadata-eval 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. +-commutative 99.9%

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

      9. *-commutative 99.9%

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

      10. div-inv 99.9%

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

      11. metadata-eval 99.9%

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

      12. *-rgt-identity 99.9%

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

      13. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -4.99999999999999954e-22 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 57.5%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. div-inv 99.1%

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

      2. pow-flip 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   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. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x} \leq -5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\frac{1}{1 + x} - \frac{1}{x} \leq 0:\\ \;\;\;\;-{x}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 3.9e+61) (+ (- 1.0 x) (/ -1.0 x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 3.9e+61) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 3.9d+61) then
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 3.9e+61) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 3.9e+61:
		tmp = (1.0 - x) + (-1.0 / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 3.9e+61)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 3.9e+61)
		tmp = (1.0 - x) + (-1.0 / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 3.9e+61], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.89999999999999987e61 < x

   1. Initial program 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. inv-pow 65.4%

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

      3. add-sqr-sqrt 22.7%

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

      4. unpow-prod-down 16.8%

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

      5. prod-diff 5.4%

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

      6. unpow-prod-down 5.3%

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

      7. add-sqr-sqrt 5.2%

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

      8. inv-pow 5.2%

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

      9. fma-define 5.2%

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

      10. *-un-lft-identity 5.2%

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

      11. +-commutative 5.2%

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

      12. distribute-neg-frac 5.2%

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

      13. metadata-eval 5.2%

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

      14. unpow-prod-down 5.3%

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

      15. add-sqr-sqrt 5.4%

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

      16. inv-pow 5.4%

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

   4. Applied egg-rr 5.4%

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

      1. fma-undefine 16.8%

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

      2. +-commutative 16.8%

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

      3. associate-+r+ 16.8%

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

      4. associate-+r+ 16.8%

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

      5. *-rgt-identity 16.8%

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

      6. metadata-eval 16.8%

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

      7. associate-*l/ 16.8%

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

      8. distribute-lft-out 16.8%

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

      9. metadata-eval 16.8%

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

      10. mul0-rgt 16.8%

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

      11. +-rgt-identity 16.8%

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

   6. Simplified 14.5%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 63.6%

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

      3. metadata-eval 63.6%

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

      4. metadata-eval 63.6%

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

      5. metadata-eval 63.6%

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

      6. associate-*r/ 63.6%

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

      7. mul0-lft 63.6%

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

   9. Simplified 63.6%

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

   if -1 < x < 3.89999999999999987e61

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 86.5%

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

      1. neg-mul-1 86.5%

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

      2. unsub-neg 86.5%

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

   5. Simplified 86.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (- 1.0 (/ 1.0 x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = 1.0 - (1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 57.9%

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

      1. *-un-lft-identity 57.9%

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

      2. inv-pow 57.9%

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

      3. add-sqr-sqrt 21.4%

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

      4. unpow-prod-down 16.4%

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

      5. prod-diff 6.8%

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

      6. unpow-prod-down 6.8%

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

      7. add-sqr-sqrt 6.7%

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

      8. inv-pow 6.7%

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

      9. fma-define 6.7%

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

      10. *-un-lft-identity 6.7%

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

      11. +-commutative 6.7%

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

      12. distribute-neg-frac 6.7%

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

      13. metadata-eval 6.7%

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

      14. unpow-prod-down 6.8%

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

      15. add-sqr-sqrt 6.8%

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

      16. inv-pow 6.8%

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

   4. Applied egg-rr 6.8%

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

      1. fma-undefine 16.4%

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

      2. +-commutative 16.4%

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

      3. associate-+r+ 16.4%

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

      4. associate-+r+ 16.4%

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

      5. *-rgt-identity 16.4%

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

      6. metadata-eval 16.4%

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

      7. associate-*l/ 16.4%

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

      8. distribute-lft-out 16.4%

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

      9. metadata-eval 16.4%

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

      10. mul0-rgt 16.4%

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

      11. +-rgt-identity 16.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 55.0%

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

      3. metadata-eval 55.0%

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

      4. metadata-eval 55.0%

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

      5. metadata-eval 55.0%

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

      6. associate-*r/ 55.0%

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

      7. mul0-lft 55.0%

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

   9. Simplified 55.0%

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

   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 98.2%

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

      1. div-sub 98.2%

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

      2. *-inverses 98.2%

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

   5. Simplified 98.2%

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

Alternative 4: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 4.5e+102) (/ -1.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 4.5e+102) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 4.5d+102) then
        tmp = (-1.0d0) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 4.5e+102) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 4.5e+102:
		tmp = -1.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 4.5e+102)
		tmp = Float64(-1.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 4.5e+102)
		tmp = -1.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 4.5e+102], N[(-1.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.50000000000000021e102 < x

   1. Initial program 69.9%

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

      1. *-un-lft-identity 69.9%

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

      2. inv-pow 69.9%

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

      3. add-sqr-sqrt 24.0%

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

      4. unpow-prod-down 17.6%

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

      5. prod-diff 5.4%

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

      6. unpow-prod-down 5.4%

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

      7. add-sqr-sqrt 5.4%

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

      8. inv-pow 5.4%

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

      9. fma-define 5.4%

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

      10. *-un-lft-identity 5.4%

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

      11. +-commutative 5.4%

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

      12. distribute-neg-frac 5.4%

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

      13. metadata-eval 5.4%

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

      14. unpow-prod-down 5.4%

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

      15. add-sqr-sqrt 5.4%

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

      16. inv-pow 5.4%

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

   4. Applied egg-rr 5.4%

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

      1. fma-undefine 17.6%

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

      2. +-commutative 17.6%

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

      3. associate-+r+ 17.6%

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

      4. associate-+r+ 17.6%

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

      5. *-rgt-identity 17.6%

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

      6. metadata-eval 17.6%

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

      7. associate-*l/ 17.6%

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

      8. distribute-lft-out 17.6%

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

      9. metadata-eval 17.6%

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

      10. mul0-rgt 17.6%

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

      11. +-rgt-identity 17.6%

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

   6. Simplified 15.2%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 68.0%

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

      3. metadata-eval 68.0%

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

      4. metadata-eval 68.0%

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

      5. metadata-eval 68.0%

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

      6. associate-*r/ 68.0%

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

      7. mul0-lft 68.0%

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

   9. Simplified 68.0%

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

   if -1 < x < 4.50000000000000021e102

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 80.6%

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

Alternative 5: 78.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. clear-num 78.1%

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

   2. frac-sub 78.9%

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

   3. *-un-lft-identity 78.9%

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

   4. div-inv 78.9%

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

   5. metadata-eval 78.9%

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

   6. *-rgt-identity 78.9%

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

   7. *-rgt-identity 78.9%

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

   8. +-commutative 78.9%

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

   9. *-commutative 78.9%

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

   10. div-inv 78.9%

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

   11. metadata-eval 78.9%

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

   12. *-rgt-identity 78.9%

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

   13. +-commutative 78.9%

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

4. Applied egg-rr 78.9%

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

Alternative 6: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Final simplification 78.1%

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

Alternative 7: 27.0% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 78.1%

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

   1. *-un-lft-identity 78.1%

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

   2. inv-pow 78.1%

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

   3. add-sqr-sqrt 34.7%

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

   4. unpow-prod-down 32.1%

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

   5. prod-diff 27.2%

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

   6. unpow-prod-down 27.2%

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

   7. add-sqr-sqrt 27.3%

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

   8. inv-pow 27.3%

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

   9. fma-define 27.3%

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

   10. *-un-lft-identity 27.3%

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

   11. +-commutative 27.3%

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

   12. distribute-neg-frac 27.3%

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

   13. metadata-eval 27.3%

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

   14. unpow-prod-down 27.2%

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

   15. add-sqr-sqrt 27.2%

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

   16. inv-pow 27.2%

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

4. Applied egg-rr 27.2%

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

   1. fma-undefine 32.1%

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

   2. +-commutative 32.1%

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

   3. associate-+r+ 32.1%

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

   4. associate-+r+ 32.1%

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

   5. *-rgt-identity 32.1%

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

   6. metadata-eval 32.1%

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

   7. associate-*l/ 32.1%

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

   8. distribute-lft-out 32.1%

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

   9. metadata-eval 32.1%

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

   10. mul0-rgt 32.1%

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

   11. +-rgt-identity 32.1%

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

6. Simplified 31.1%

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

7. Taylor expanded in x around -inf 0.0%

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

   1. unpow2 0.0%

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

   2. rem-square-sqrt 29.7%

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

   3. metadata-eval 29.7%

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

   4. metadata-eval 29.7%

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

   5. metadata-eval 29.7%

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

   6. associate-*r/ 29.7%

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

   7. mul0-lft 29.7%

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

9. Simplified 29.7%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))