Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

Alternatives: 6

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{x - y} + 2\\ \frac{t_0}{t_0 \cdot \frac{x - y}{x + y}} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (+ x y) (- x y)) 2.0)))
   (/ t_0 (* t_0 (/ (- x y) (+ x y))))))

C

double code(double x, double y) {
	double t_0 = ((x + y) / (x - y)) + 2.0;
	return t_0 / (t_0 * ((x - y) / (x + y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = ((x + y) / (x - y)) + 2.0d0
    code = t_0 / (t_0 * ((x - y) / (x + y)))
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x + y) / (x - y)) + 2.0;
	return t_0 / (t_0 * ((x - y) / (x + y)));
}

Python

def code(x, y):
	t_0 = ((x + y) / (x - y)) + 2.0
	return t_0 / (t_0 * ((x - y) / (x + y)))

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x + y) / Float64(x - y)) + 2.0)
	return Float64(t_0 / Float64(t_0 * Float64(Float64(x - y) / Float64(x + y))))
end

MATLAB

function tmp = code(x, y)
	t_0 = ((x + y) / (x - y)) + 2.0;
	tmp = t_0 / (t_0 * ((x - y) / (x + y)));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(t$95$0 / N[(t$95$0 * N[(N[(x - y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + y}{x - y}\right)\right)} \]

4. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + y}{x - y}\right)\right)} \]
5. Step-by-step derivation

   1. expm1-udef 98.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} - 1} \]

   2. flip-- 98.0%

      \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} \cdot e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1}} \]

   3. log1p-udef 98.0%

      \[\leadsto \frac{e^{\color{blue}{\log \left(1 + \frac{x + y}{x - y}\right)}} \cdot e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1} \]

   4. rem-exp-log 98.0%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{x + y}{x - y}\right)} \cdot e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1} \]

   5. +-commutative 98.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{x + y}{x - y} + 1\right)} \cdot e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1} \]

   6. log1p-udef 98.0%

      \[\leadsto \frac{\left(\frac{x + y}{x - y} + 1\right) \cdot e^{\color{blue}{\log \left(1 + \frac{x + y}{x - y}\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1} \]

   7. rem-exp-log 98.0%

      \[\leadsto \frac{\left(\frac{x + y}{x - y} + 1\right) \cdot \color{blue}{\left(1 + \frac{x + y}{x - y}\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1} \]

   8. +-commutative 98.0%

      \[\leadsto \frac{\left(\frac{x + y}{x - y} + 1\right) \cdot \color{blue}{\left(\frac{x + y}{x - y} + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1} \]

   9. metadata-eval 98.0%

      \[\leadsto \frac{\left(\frac{x + y}{x - y} + 1\right) \cdot \left(\frac{x + y}{x - y} + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + 1} \]

   10. log1p-udef 98.0%

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

   11. rem-exp-log 100.0%

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

   12. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. div-inv 100.0%

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

   2. difference-of-sqr-1 100.0%

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

   3. associate-*l* 99.9%

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

   4. associate-+l+ 99.9%

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

   5. metadata-eval 99.9%

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

   6. add-exp-log 98.0%

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

   7. +-commutative 98.0%

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

   8. log1p-udef 98.0%

      \[\leadsto \left(\frac{x + y}{x - y} + 2\right) \cdot \left(\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)}} - 1\right) \cdot \frac{1}{\left(\frac{x + y}{x - y} + 1\right) + 1}\right) \]

   9. expm1-udef 98.0%

      \[\leadsto \left(\frac{x + y}{x - y} + 2\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + y}{x - y}\right)\right)} \cdot \frac{1}{\left(\frac{x + y}{x - y} + 1\right) + 1}\right) \]

   10. expm1-log1p-u 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

   2. div-inv 100.0%

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

   3. associate-/l* 100.0%

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

   4. div-inv 100.0%

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

   5. clear-num 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

    \[\leadsto \frac{\frac{x + y}{x - y} + 2}{\left(\frac{x + y}{x - y} + 2\right) \cdot \frac{x - y}{x + y}} \]
12. Add Preprocessing

Alternative 2: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-24} \lor \neg \left(x \leq 1.26 \cdot 10^{-92}\right) \land \left(x \leq 1.2 \cdot 10^{+25} \lor \neg \left(x \leq 2.22 \cdot 10^{+51}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.4e-24)
         (and (not (<= x 1.26e-92)) (or (<= x 1.2e+25) (not (<= x 2.22e+51)))))
   (+ 1.0 (* 2.0 (/ y x)))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.4e-24) || (!(x <= 1.26e-92) && ((x <= 1.2e+25) || !(x <= 2.22e+51)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} 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 ((x <= (-6.4d-24)) .or. (.not. (x <= 1.26d-92)) .and. (x <= 1.2d+25) .or. (.not. (x <= 2.22d+51))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6.4e-24) || (!(x <= 1.26e-92) && ((x <= 1.2e+25) || !(x <= 2.22e+51)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6.4e-24) or (not (x <= 1.26e-92) and ((x <= 1.2e+25) or not (x <= 2.22e+51))):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.4e-24) || (!(x <= 1.26e-92) && ((x <= 1.2e+25) || !(x <= 2.22e+51))))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6.4e-24) || (~((x <= 1.26e-92)) && ((x <= 1.2e+25) || ~((x <= 2.22e+51)))))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.4e-24], And[N[Not[LessEqual[x, 1.26e-92]], $MachinePrecision], Or[LessEqual[x, 1.2e+25], N[Not[LessEqual[x, 2.22e+51]], $MachinePrecision]]]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -6.40000000000000025e-24 or 1.26000000000000006e-92 < x < 1.19999999999999998e25 or 2.2199999999999999e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.0%

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

   if -6.40000000000000025e-24 < x < 1.26000000000000006e-92 or 1.19999999999999998e25 < x < 2.2199999999999999e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-24} \lor \neg \left(x \leq 1.26 \cdot 10^{-92}\right) \land \left(x \leq 1.2 \cdot 10^{+25} \lor \neg \left(x \leq 2.22 \cdot 10^{+51}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-25} \lor \neg \left(x \leq 5.8 \cdot 10^{-93}\right) \land \left(x \leq 4.7 \cdot 10^{+24} \lor \neg \left(x \leq 3.7 \cdot 10^{+52}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4e-25)
         (and (not (<= x 5.8e-93)) (or (<= x 4.7e+24) (not (<= x 3.7e+52)))))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4e-25) || (!(x <= 5.8e-93) && ((x <= 4.7e+24) || !(x <= 3.7e+52)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -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 ((x <= (-4d-25)) .or. (.not. (x <= 5.8d-93)) .and. (x <= 4.7d+24) .or. (.not. (x <= 3.7d+52))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4e-25) || (!(x <= 5.8e-93) && ((x <= 4.7e+24) || !(x <= 3.7e+52)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4e-25) or (not (x <= 5.8e-93) and ((x <= 4.7e+24) or not (x <= 3.7e+52))):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = (-2.0 * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4e-25) || (!(x <= 5.8e-93) && ((x <= 4.7e+24) || !(x <= 3.7e+52))))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4e-25) || (~((x <= 5.8e-93)) && ((x <= 4.7e+24) || ~((x <= 3.7e+52)))))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = (-2.0 * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4e-25], And[N[Not[LessEqual[x, 5.8e-93]], $MachinePrecision], Or[LessEqual[x, 4.7e+24], N[Not[LessEqual[x, 3.7e+52]], $MachinePrecision]]]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000015e-25 or 5.7999999999999997e-93 < x < 4.7e24 or 3.7e52 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.0%

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

   if -4.00000000000000015e-25 < x < 5.7999999999999997e-93 or 4.7e24 < x < 3.7e52

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 83.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-25} \lor \neg \left(x \leq 5.8 \cdot 10^{-93}\right) \land \left(x \leq 4.7 \cdot 10^{+24} \lor \neg \left(x \leq 3.7 \cdot 10^{+52}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-92}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-30)
   1.0
   (if (<= x 1.55e-92)
     -1.0
     (if (<= x 6.1e+24) 1.0 (if (<= x 4.1e+52) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-30) {
		tmp = 1.0;
	} else if (x <= 1.55e-92) {
		tmp = -1.0;
	} else if (x <= 6.1e+24) {
		tmp = 1.0;
	} else if (x <= 4.1e+52) {
		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 (x <= (-1d-30)) then
        tmp = 1.0d0
    else if (x <= 1.55d-92) then
        tmp = -1.0d0
    else if (x <= 6.1d+24) then
        tmp = 1.0d0
    else if (x <= 4.1d+52) 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 (x <= -1e-30) {
		tmp = 1.0;
	} else if (x <= 1.55e-92) {
		tmp = -1.0;
	} else if (x <= 6.1e+24) {
		tmp = 1.0;
	} else if (x <= 4.1e+52) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-30:
		tmp = 1.0
	elif x <= 1.55e-92:
		tmp = -1.0
	elif x <= 6.1e+24:
		tmp = 1.0
	elif x <= 4.1e+52:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-30)
		tmp = 1.0;
	elseif (x <= 1.55e-92)
		tmp = -1.0;
	elseif (x <= 6.1e+24)
		tmp = 1.0;
	elseif (x <= 4.1e+52)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-30)
		tmp = 1.0;
	elseif (x <= 1.55e-92)
		tmp = -1.0;
	elseif (x <= 6.1e+24)
		tmp = 1.0;
	elseif (x <= 4.1e+52)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-30], 1.0, If[LessEqual[x, 1.55e-92], -1.0, If[LessEqual[x, 6.1e+24], 1.0, If[LessEqual[x, 4.1e+52], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-30 or 1.55e-92 < x < 6.10000000000000006e24 or 4.1e52 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 75.8%

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

   if -1e-30 < x < 1.55e-92 or 6.10000000000000006e24 < x < 4.1e52

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 83.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-92}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 6: 49.9% 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 48.7%

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

4. Final simplification 48.7%

   \[\leadsto -1 \]
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 2024024 
(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)))