Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x + y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(x - y))
end

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x + y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(x - y))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

4. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

   2. clear-num 100.0%

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

   3. flip3-- 35.4%

      \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

   4. associate-/r/ 35.4%

      \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(x \cdot x + \left(y \cdot y + x \cdot y\right)\right)} \]

   5. unpow2 35.4%

      \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(\color{blue}{{x}^{2}} + \left(y \cdot y + x \cdot y\right)\right) \]

   6. distribute-rgt-in 35.4%

      \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left({x}^{2} + \color{blue}{y \cdot \left(y + x\right)}\right) \]

   7. +-commutative 35.4%

      \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left({x}^{2} + y \cdot \color{blue}{\left(x + y\right)}\right) \]

   8. +-commutative 35.4%

      \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\left(y \cdot \left(x + y\right) + {x}^{2}\right)} \]

   9. fma-udef 35.4%

      \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)} \]

   10. associate-/r/ 35.4%

       \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   11. clear-num 35.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}{x + y}}} \]

   12. associate-/l/ 34.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{\left(x + y\right) \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   13. fma-udef 34.9%

       \[\leadsto \frac{1}{\frac{{x}^{3} - {y}^{3}}{\left(x + y\right) \cdot \color{blue}{\left(y \cdot \left(x + y\right) + {x}^{2}\right)}}} \]

   14. +-commutative 34.9%

       \[\leadsto \frac{1}{\frac{{x}^{3} - {y}^{3}}{\left(x + y\right) \cdot \color{blue}{\left({x}^{2} + y \cdot \left(x + y\right)\right)}}} \]

   15. +-commutative 34.9%

       \[\leadsto \frac{1}{\frac{{x}^{3} - {y}^{3}}{\left(x + y\right) \cdot \left({x}^{2} + y \cdot \color{blue}{\left(y + x\right)}\right)}} \]

   16. distribute-rgt-in 34.9%

       \[\leadsto \frac{1}{\frac{{x}^{3} - {y}^{3}}{\left(x + y\right) \cdot \left({x}^{2} + \color{blue}{\left(y \cdot y + x \cdot y\right)}\right)}} \]

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+103} \lor \neg \left(y \leq 1.3 \cdot 10^{-194}\right) \land \left(y \leq 9.8 \cdot 10^{-156} \lor \neg \left(y \leq 420000000\right)\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.2e+103)
         (and (not (<= y 1.3e-194))
              (or (<= y 9.8e-156) (not (<= y 420000000.0)))))
   (- -1.0 (/ x y))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+103) || (!(y <= 1.3e-194) && ((y <= 9.8e-156) || !(y <= 420000000.0)))) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.2d+103)) .or. (.not. (y <= 1.3d-194)) .and. (y <= 9.8d-156) .or. (.not. (y <= 420000000.0d0))) then
        tmp = (-1.0d0) - (x / y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+103) || (!(y <= 1.3e-194) && ((y <= 9.8e-156) || !(y <= 420000000.0)))) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.2e+103) or (not (y <= 1.3e-194) and ((y <= 9.8e-156) or not (y <= 420000000.0))):
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.2e+103) || (!(y <= 1.3e-194) && ((y <= 9.8e-156) || !(y <= 420000000.0))))
		tmp = Float64(-1.0 - Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.2e+103) || (~((y <= 1.3e-194)) && ((y <= 9.8e-156) || ~((y <= 420000000.0)))))
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.2e+103], And[N[Not[LessEqual[y, 1.3e-194]], $MachinePrecision], Or[LessEqual[y, 9.8e-156], N[Not[LessEqual[y, 420000000.0]], $MachinePrecision]]]], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -4.2000000000000003e103 or 1.30000000000000001e-194 < y < 9.79999999999999902e-156 or 4.2e8 < y

   1. Initial program 100.0%

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

      1. flip3-- 11.0%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      2. associate-/r/ 11.0%

         \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(x \cdot x + \left(y \cdot y + x \cdot y\right)\right)} \]

      3. +-commutative 11.0%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + x \cdot y\right) + x \cdot x\right)} \]

      4. distribute-rgt-out 11.0%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(\color{blue}{y \cdot \left(y + x\right)} + x \cdot x\right) \]

      5. +-commutative 11.0%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(y \cdot \color{blue}{\left(x + y\right)} + x \cdot x\right) \]

      6. fma-def 11.0%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\mathsf{fma}\left(y, x + y, x \cdot x\right)} \]

      7. pow2 11.0%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, \color{blue}{{x}^{2}}\right) \]

   4. Applied egg-rr 11.0%

      \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 10.3%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)}{{x}^{3} - {y}^{3}}} \]

      2. associate-/l* 11.0%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   6. Simplified 11.0%

      \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   7. Taylor expanded in x around 0 78.9%

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

      1. neg-mul-1 78.9%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in x around 0 78.9%

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

       1. fma-neg 78.9%

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

       2. metadata-eval 78.9%

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

       3. *-lft-identity 78.9%

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

       4. metadata-eval 78.9%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot -1\right) \cdot y}, -1\right) \]

       7. rem-square-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot y}, -1\right) \]

       8. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \cdot y}, -1\right) \]

       9. pow-sqr 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\color{blue}{{\left(\sqrt{-1}\right)}^{\left(2 \cdot 2\right)}} \cdot y}, -1\right) \]

       10. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{{\left(\sqrt{-1}\right)}^{\color{blue}{4}} \cdot y}, -1\right) \]

       11. *-commutative 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\color{blue}{y \cdot {\left(\sqrt{-1}\right)}^{4}}}, -1\right) \]

       12. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \color{blue}{\frac{1}{-1}}\right) \]

       13. rem-square-sqrt 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right) \]

       14. unpow2 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \frac{1}{\color{blue}{{\left(\sqrt{-1}\right)}^{2}}}\right) \]

       15. fma-def 0.0%

           \[\leadsto \color{blue}{-1 \cdot \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}} \]

       16. +-commutative 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} + -1 \cdot \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}} \]

       17. mul-1-neg 0.0%

           \[\leadsto \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \color{blue}{\left(-\frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}\right)} \]

       18. sub-neg 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}} \]

       19. unpow2 0.0%

           \[\leadsto \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       20. rem-square-sqrt 0.0%

           \[\leadsto \frac{1}{\color{blue}{-1}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       21. metadata-eval 0.0%

           \[\leadsto \color{blue}{-1} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       22. *-commutative 0.0%

           \[\leadsto -1 - \frac{x}{\color{blue}{{\left(\sqrt{-1}\right)}^{4} \cdot y}} \]

   12. Simplified 78.9%

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

   if -4.2000000000000003e103 < y < 1.30000000000000001e-194 or 9.79999999999999902e-156 < y < 4.2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 77.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+103} \lor \neg \left(y \leq 1.3 \cdot 10^{-194}\right) \land \left(y \leq 9.8 \cdot 10^{-156} \lor \neg \left(y \leq 420000000\right)\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.1 \cdot 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+103)
   -1.0
   (if (<= y 1.15e-179)
     1.0
     (if (<= y 8.1e-156) -1.0 (if (<= y 2000000000.0) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+103) {
		tmp = -1.0;
	} else if (y <= 1.15e-179) {
		tmp = 1.0;
	} else if (y <= 8.1e-156) {
		tmp = -1.0;
	} else if (y <= 2000000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.2d+103)) then
        tmp = -1.0d0
    else if (y <= 1.15d-179) then
        tmp = 1.0d0
    else if (y <= 8.1d-156) then
        tmp = -1.0d0
    else if (y <= 2000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+103) {
		tmp = -1.0;
	} else if (y <= 1.15e-179) {
		tmp = 1.0;
	} else if (y <= 8.1e-156) {
		tmp = -1.0;
	} else if (y <= 2000000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+103:
		tmp = -1.0
	elif y <= 1.15e-179:
		tmp = 1.0
	elif y <= 8.1e-156:
		tmp = -1.0
	elif y <= 2000000000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+103)
		tmp = -1.0;
	elseif (y <= 1.15e-179)
		tmp = 1.0;
	elseif (y <= 8.1e-156)
		tmp = -1.0;
	elseif (y <= 2000000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+103)
		tmp = -1.0;
	elseif (y <= 1.15e-179)
		tmp = 1.0;
	elseif (y <= 8.1e-156)
		tmp = -1.0;
	elseif (y <= 2000000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+103], -1.0, If[LessEqual[y, 1.15e-179], 1.0, If[LessEqual[y, 8.1e-156], -1.0, If[LessEqual[y, 2000000000.0], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2000000000000003e103 or 1.14999999999999994e-179 < y < 8.0999999999999998e-156 or 2e9 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.8%

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

   if -4.2000000000000003e103 < y < 1.14999999999999994e-179 or 8.0999999999999998e-156 < y < 2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 76.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.1 \cdot 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+103} \lor \neg \left(y \leq 2000000000000\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+103) (not (<= y 2000000000000.0)))
   (- -1.0 (/ x y))
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+103) || !(y <= 2000000000000.0)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8d+103)) .or. (.not. (y <= 2000000000000.0d0))) then
        tmp = (-1.0d0) - (x / y)
    else
        tmp = 1.0d0 + (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e+103) || !(y <= 2000000000000.0)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e+103) or not (y <= 2000000000000.0):
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0 + (2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e+103) || !(y <= 2000000000000.0))
		tmp = Float64(-1.0 - Float64(x / y));
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+103) || ~((y <= 2000000000000.0)))
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0 + (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e+103], N[Not[LessEqual[y, 2000000000000.0]], $MachinePrecision]], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e103 or 2e12 < y

   1. Initial program 100.0%

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

      1. flip3-- 10.9%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      2. associate-/r/ 10.9%

         \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(x \cdot x + \left(y \cdot y + x \cdot y\right)\right)} \]

      3. +-commutative 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + x \cdot y\right) + x \cdot x\right)} \]

      4. distribute-rgt-out 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(\color{blue}{y \cdot \left(y + x\right)} + x \cdot x\right) \]

      5. +-commutative 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(y \cdot \color{blue}{\left(x + y\right)} + x \cdot x\right) \]

      6. fma-def 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\mathsf{fma}\left(y, x + y, x \cdot x\right)} \]

      7. pow2 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, \color{blue}{{x}^{2}}\right) \]

   4. Applied egg-rr 10.9%

      \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 10.3%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)}{{x}^{3} - {y}^{3}}} \]

      2. associate-/l* 10.9%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   6. Simplified 10.9%

      \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   7. Taylor expanded in x around 0 79.7%

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

      1. neg-mul-1 79.7%

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

   9. Simplified 79.7%

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

   10. Taylor expanded in x around 0 79.7%

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

       1. fma-neg 79.7%

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

       2. metadata-eval 79.7%

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

       3. *-lft-identity 79.7%

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

       4. metadata-eval 79.7%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot -1\right) \cdot y}, -1\right) \]

       7. rem-square-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot y}, -1\right) \]

       8. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \cdot y}, -1\right) \]

       9. pow-sqr 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\color{blue}{{\left(\sqrt{-1}\right)}^{\left(2 \cdot 2\right)}} \cdot y}, -1\right) \]

       10. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{{\left(\sqrt{-1}\right)}^{\color{blue}{4}} \cdot y}, -1\right) \]

       11. *-commutative 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\color{blue}{y \cdot {\left(\sqrt{-1}\right)}^{4}}}, -1\right) \]

       12. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \color{blue}{\frac{1}{-1}}\right) \]

       13. rem-square-sqrt 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right) \]

       14. unpow2 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \frac{1}{\color{blue}{{\left(\sqrt{-1}\right)}^{2}}}\right) \]

       15. fma-def 0.0%

           \[\leadsto \color{blue}{-1 \cdot \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}} \]

       16. +-commutative 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} + -1 \cdot \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}} \]

       17. mul-1-neg 0.0%

           \[\leadsto \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \color{blue}{\left(-\frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}\right)} \]

       18. sub-neg 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}} \]

       19. unpow2 0.0%

           \[\leadsto \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       20. rem-square-sqrt 0.0%

           \[\leadsto \frac{1}{\color{blue}{-1}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       21. metadata-eval 0.0%

           \[\leadsto \color{blue}{-1} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       22. *-commutative 0.0%

           \[\leadsto -1 - \frac{x}{\color{blue}{{\left(\sqrt{-1}\right)}^{4} \cdot y}} \]

   12. Simplified 79.7%

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

   if -8e103 < y < 2e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 74.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+103} \lor \neg \left(y \leq 2000000000000\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+103} \lor \neg \left(y \leq 12500000\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.2e+103) (not (<= y 12500000.0)))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+103) || !(y <= 12500000.0)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.2d+103)) .or. (.not. (y <= 12500000.0d0))) then
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+103) || !(y <= 12500000.0)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.2e+103) or not (y <= 12500000.0):
		tmp = (-2.0 * (x / y)) + -1.0
	else:
		tmp = 1.0 + (2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.2e+103) || !(y <= 12500000.0))
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.2e+103) || ~((y <= 12500000.0)))
		tmp = (-2.0 * (x / y)) + -1.0;
	else
		tmp = 1.0 + (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.2e+103], N[Not[LessEqual[y, 12500000.0]], $MachinePrecision]], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2000000000000003e103 or 1.25e7 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.1%

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

   if -4.2000000000000003e103 < y < 1.25e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 74.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+103} \lor \neg \left(y \leq 12500000\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+103} \lor \neg \left(y \leq 1450000000\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.5e+103) (not (<= y 1450000000.0)))
   (- -1.0 (/ x y))
   (/ (+ x y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e+103) || !(y <= 1450000000.0)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = (x + y) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.5d+103)) .or. (.not. (y <= 1450000000.0d0))) then
        tmp = (-1.0d0) - (x / y)
    else
        tmp = (x + y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e+103) || !(y <= 1450000000.0)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = (x + y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.5e+103) or not (y <= 1450000000.0):
		tmp = -1.0 - (x / y)
	else:
		tmp = (x + y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.5e+103) || !(y <= 1450000000.0))
		tmp = Float64(-1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x + y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.5e+103) || ~((y <= 1450000000.0)))
		tmp = -1.0 - (x / y);
	else
		tmp = (x + y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.5e+103], N[Not[LessEqual[y, 1450000000.0]], $MachinePrecision]], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.49999999999999922e103 or 1.45e9 < y

   1. Initial program 100.0%

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

      1. flip3-- 10.9%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      2. associate-/r/ 10.9%

         \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(x \cdot x + \left(y \cdot y + x \cdot y\right)\right)} \]

      3. +-commutative 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + x \cdot y\right) + x \cdot x\right)} \]

      4. distribute-rgt-out 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(\color{blue}{y \cdot \left(y + x\right)} + x \cdot x\right) \]

      5. +-commutative 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(y \cdot \color{blue}{\left(x + y\right)} + x \cdot x\right) \]

      6. fma-def 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\mathsf{fma}\left(y, x + y, x \cdot x\right)} \]

      7. pow2 10.9%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, \color{blue}{{x}^{2}}\right) \]

   4. Applied egg-rr 10.9%

      \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 10.3%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)}{{x}^{3} - {y}^{3}}} \]

      2. associate-/l* 10.9%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   6. Simplified 10.9%

      \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   7. Taylor expanded in x around 0 79.7%

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

      1. neg-mul-1 79.7%

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

   9. Simplified 79.7%

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

   10. Taylor expanded in x around 0 79.7%

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

       1. fma-neg 79.7%

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

       2. metadata-eval 79.7%

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

       3. *-lft-identity 79.7%

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

       4. metadata-eval 79.7%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot -1\right) \cdot y}, -1\right) \]

       7. rem-square-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot y}, -1\right) \]

       8. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \cdot y}, -1\right) \]

       9. pow-sqr 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\color{blue}{{\left(\sqrt{-1}\right)}^{\left(2 \cdot 2\right)}} \cdot y}, -1\right) \]

       10. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{{\left(\sqrt{-1}\right)}^{\color{blue}{4}} \cdot y}, -1\right) \]

       11. *-commutative 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\color{blue}{y \cdot {\left(\sqrt{-1}\right)}^{4}}}, -1\right) \]

       12. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \color{blue}{\frac{1}{-1}}\right) \]

       13. rem-square-sqrt 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right) \]

       14. unpow2 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}, \frac{1}{\color{blue}{{\left(\sqrt{-1}\right)}^{2}}}\right) \]

       15. fma-def 0.0%

           \[\leadsto \color{blue}{-1 \cdot \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}} \]

       16. +-commutative 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} + -1 \cdot \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}} \]

       17. mul-1-neg 0.0%

           \[\leadsto \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \color{blue}{\left(-\frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}\right)} \]

       18. sub-neg 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}}} \]

       19. unpow2 0.0%

           \[\leadsto \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       20. rem-square-sqrt 0.0%

           \[\leadsto \frac{1}{\color{blue}{-1}} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       21. metadata-eval 0.0%

           \[\leadsto \color{blue}{-1} - \frac{x}{y \cdot {\left(\sqrt{-1}\right)}^{4}} \]

       22. *-commutative 0.0%

           \[\leadsto -1 - \frac{x}{\color{blue}{{\left(\sqrt{-1}\right)}^{4} \cdot y}} \]

   12. Simplified 79.7%

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

   if -7.49999999999999922e103 < y < 1.45e9

   1. Initial program 100.0%

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

      1. flip3-- 50.3%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      2. associate-/r/ 50.3%

         \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(x \cdot x + \left(y \cdot y + x \cdot y\right)\right)} \]

      3. +-commutative 50.3%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + x \cdot y\right) + x \cdot x\right)} \]

      4. distribute-rgt-out 50.3%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(\color{blue}{y \cdot \left(y + x\right)} + x \cdot x\right) \]

      5. +-commutative 50.3%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \left(y \cdot \color{blue}{\left(x + y\right)} + x \cdot x\right) \]

      6. fma-def 50.3%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\mathsf{fma}\left(y, x + y, x \cdot x\right)} \]

      7. pow2 50.3%

         \[\leadsto \frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, \color{blue}{{x}^{2}}\right) \]

   4. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{\frac{x + y}{{x}^{3} - {y}^{3}} \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 50.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)}{{x}^{3} - {y}^{3}}} \]

      2. associate-/l* 50.3%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   6. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{x + y}{\frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)}}} \]

   7. Taylor expanded in x around inf 74.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+103} \lor \neg \left(y \leq 1450000000\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x + y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(x - y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 8: 49.8% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 46.0%

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

4. Final simplification 46.0%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (+ x y) (- x y)))