Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 99.9%

Time: 15.2s

Alternatives: 7

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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{x - y}\\ t_1 := 1 + t\_0\\ \frac{1}{2 + t\_0} \cdot \left(-1 + \left(t\_1 + t\_0 \cdot t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- x y))) (t_1 (+ 1.0 t_0)))
   (* (/ 1.0 (+ 2.0 t_0)) (+ -1.0 (+ t_1 (* t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (x - y);
	double t_1 = 1.0 + t_0;
	return (1.0 / (2.0 + t_0)) * (-1.0 + (t_1 + (t_0 * t_1)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (y + x) / (x - y)
    t_1 = 1.0d0 + t_0
    code = (1.0d0 / (2.0d0 + t_0)) * ((-1.0d0) + (t_1 + (t_0 * t_1)))
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) / (x - y);
	double t_1 = 1.0 + t_0;
	return (1.0 / (2.0 + t_0)) * (-1.0 + (t_1 + (t_0 * t_1)));
}

Python

def code(x, y):
	t_0 = (y + x) / (x - y)
	t_1 = 1.0 + t_0
	return (1.0 / (2.0 + t_0)) * (-1.0 + (t_1 + (t_0 * t_1)))

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(x - y))
	t_1 = Float64(1.0 + t_0)
	return Float64(Float64(1.0 / Float64(2.0 + t_0)) * Float64(-1.0 + Float64(t_1 + Float64(t_0 * t_1))))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y + x) / (x - y);
	t_1 = 1.0 + t_0;
	tmp = (1.0 / (2.0 + t_0)) * (-1.0 + (t_1 + (t_0 * t_1)));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, N[(N[(1.0 / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(t$95$1 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 98.8%

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

4. Applied egg-rr 98.8%

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

   1. expm1-define 98.7%

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

   2. flip-- 98.7%

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

   3. div-inv 98.7%

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

   4. metadata-eval 98.7%

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

   5. sub-neg 98.7%

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

   6. pow2 98.7%

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

   7. log1p-undefine 98.8%

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

   8. rem-exp-log 98.8%

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

   9. metadata-eval 98.8%

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

   10. +-commutative 98.8%

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

   11. log1p-undefine 98.8%

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

6. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-+r+ 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

   5. +-commutative 99.9%

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

   6. +-commutative 99.9%

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

8. Simplified 99.9%

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

   1. +-commutative 99.9%

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

   2. +-commutative 99.9%

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

   3. pow2 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-rgt-in 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. +-commutative 99.9%

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

   8. +-commutative 99.9%

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

   9. +-commutative 99.9%

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

   10. +-commutative 99.9%

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

   11. +-commutative 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \frac{1}{2 + \frac{y + x}{x - y}} \cdot \left(-1 + \left(\left(1 + \frac{y + x}{x - y}\right) + \frac{y + x}{x - y} \cdot \left(1 + \frac{y + x}{x - y}\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{x - y}\\ t_1 := 2 + t\_0\\ \frac{1}{t\_1} \cdot \left(t\_0 \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- x y))) (t_1 (+ 2.0 t_0)))
   (* (/ 1.0 t_1) (* t_0 t_1))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (x - y);
	double t_1 = 2.0 + t_0;
	return (1.0 / t_1) * (t_0 * t_1);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (y + x) / (x - y)
    t_1 = 2.0d0 + t_0
    code = (1.0d0 / t_1) * (t_0 * t_1)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) / (x - y);
	double t_1 = 2.0 + t_0;
	return (1.0 / t_1) * (t_0 * t_1);
}

Python

def code(x, y):
	t_0 = (y + x) / (x - y)
	t_1 = 2.0 + t_0
	return (1.0 / t_1) * (t_0 * t_1)

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(x - y))
	t_1 = Float64(2.0 + t_0)
	return Float64(Float64(1.0 / t_1) * Float64(t_0 * t_1))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y + x) / (x - y);
	t_1 = 2.0 + t_0;
	tmp = (1.0 / t_1) * (t_0 * t_1);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 98.8%

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

4. Applied egg-rr 98.8%

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

   1. expm1-define 98.7%

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

   2. flip-- 98.7%

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

   3. div-inv 98.7%

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

   4. metadata-eval 98.7%

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

   5. sub-neg 98.7%

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

   6. pow2 98.7%

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

   7. log1p-undefine 98.8%

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

   8. rem-exp-log 98.8%

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

   9. metadata-eval 98.8%

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

   10. +-commutative 98.8%

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

   11. log1p-undefine 98.8%

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

6. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-+r+ 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

   5. +-commutative 99.9%

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

   6. +-commutative 99.9%

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

8. Simplified 99.9%

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

   1. +-commutative 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. pow2 99.9%

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

   5. difference-of-sqr--1 99.9%

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

   6. associate-+l+ 99.9%

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

   7. +-commutative 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. associate--l+ 99.9%

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

   11. metadata-eval 99.9%

       \[\leadsto \frac{1}{2 + \frac{y + x}{x - y}} \cdot \left(\left(2 + \frac{y + x}{x - y}\right) \cdot \left(\frac{x + y}{x - y} + \color{blue}{0}\right)\right) \]

   12. +-rgt-identity 99.9%

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

   13. +-commutative 99.9%

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

10. Applied egg-rr 99.9%

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

    1. *-commutative 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

    \[\leadsto \frac{1}{2 + \frac{y + x}{x - y}} \cdot \left(\frac{y + x}{x - y} \cdot \left(2 + \frac{y + x}{x - y}\right)\right) \]
14. Add Preprocessing

Alternative 3: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-37}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ y x)))))
   (if (<= x -1.02e+55)
     t_0
     (if (<= x -1.15e+41)
       -1.0
       (if (<= x -1.5e-18) 1.0 (if (<= x 1.8e-37) -1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -1.02e+55) {
		tmp = t_0;
	} else if (x <= -1.15e+41) {
		tmp = -1.0;
	} else if (x <= -1.5e-18) {
		tmp = 1.0;
	} else if (x <= 1.8e-37) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 * (y / x))
    if (x <= (-1.02d+55)) then
        tmp = t_0
    else if (x <= (-1.15d+41)) then
        tmp = -1.0d0
    else if (x <= (-1.5d-18)) then
        tmp = 1.0d0
    else if (x <= 1.8d-37) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -1.02e+55) {
		tmp = t_0;
	} else if (x <= -1.15e+41) {
		tmp = -1.0;
	} else if (x <= -1.5e-18) {
		tmp = 1.0;
	} else if (x <= 1.8e-37) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (2.0 * (y / x))
	tmp = 0
	if x <= -1.02e+55:
		tmp = t_0
	elif x <= -1.15e+41:
		tmp = -1.0
	elif x <= -1.5e-18:
		tmp = 1.0
	elif x <= 1.8e-37:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(y / x)))
	tmp = 0.0
	if (x <= -1.02e+55)
		tmp = t_0;
	elseif (x <= -1.15e+41)
		tmp = -1.0;
	elseif (x <= -1.5e-18)
		tmp = 1.0;
	elseif (x <= 1.8e-37)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (2.0 * (y / x));
	tmp = 0.0;
	if (x <= -1.02e+55)
		tmp = t_0;
	elseif (x <= -1.15e+41)
		tmp = -1.0;
	elseif (x <= -1.5e-18)
		tmp = 1.0;
	elseif (x <= 1.8e-37)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.02e+55], t$95$0, If[LessEqual[x, -1.15e+41], -1.0, If[LessEqual[x, -1.5e-18], 1.0, If[LessEqual[x, 1.8e-37], -1.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.02000000000000002e55 or 1.80000000000000004e-37 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 79.4%

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

   if -1.02000000000000002e55 < x < -1.1499999999999999e41 or -1.49999999999999991e-18 < x < 1.80000000000000004e-37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.4%

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

   if -1.1499999999999999e41 < x < -1.49999999999999991e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 78.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+55}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-37}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+71}:\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ y x)))))
   (if (<= x -7.2e+55)
     t_0
     (if (<= x -1.2e+39)
       -1.0
       (if (<= x -1.65e-18)
         1.0
         (if (<= x 3.3e+71) (+ -1.0 (* -2.0 (/ x y))) t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -7.2e+55) {
		tmp = t_0;
	} else if (x <= -1.2e+39) {
		tmp = -1.0;
	} else if (x <= -1.65e-18) {
		tmp = 1.0;
	} else if (x <= 3.3e+71) {
		tmp = -1.0 + (-2.0 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 * (y / x))
    if (x <= (-7.2d+55)) then
        tmp = t_0
    else if (x <= (-1.2d+39)) then
        tmp = -1.0d0
    else if (x <= (-1.65d-18)) then
        tmp = 1.0d0
    else if (x <= 3.3d+71) then
        tmp = (-1.0d0) + ((-2.0d0) * (x / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -7.2e+55) {
		tmp = t_0;
	} else if (x <= -1.2e+39) {
		tmp = -1.0;
	} else if (x <= -1.65e-18) {
		tmp = 1.0;
	} else if (x <= 3.3e+71) {
		tmp = -1.0 + (-2.0 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (2.0 * (y / x))
	tmp = 0
	if x <= -7.2e+55:
		tmp = t_0
	elif x <= -1.2e+39:
		tmp = -1.0
	elif x <= -1.65e-18:
		tmp = 1.0
	elif x <= 3.3e+71:
		tmp = -1.0 + (-2.0 * (x / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(y / x)))
	tmp = 0.0
	if (x <= -7.2e+55)
		tmp = t_0;
	elseif (x <= -1.2e+39)
		tmp = -1.0;
	elseif (x <= -1.65e-18)
		tmp = 1.0;
	elseif (x <= 3.3e+71)
		tmp = Float64(-1.0 + Float64(-2.0 * Float64(x / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (2.0 * (y / x));
	tmp = 0.0;
	if (x <= -7.2e+55)
		tmp = t_0;
	elseif (x <= -1.2e+39)
		tmp = -1.0;
	elseif (x <= -1.65e-18)
		tmp = 1.0;
	elseif (x <= 3.3e+71)
		tmp = -1.0 + (-2.0 * (x / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e+55], t$95$0, If[LessEqual[x, -1.2e+39], -1.0, If[LessEqual[x, -1.65e-18], 1.0, If[LessEqual[x, 3.3e+71], N[(-1.0 + N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.19999999999999975e55 or 3.2999999999999998e71 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 85.6%

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

   if -7.19999999999999975e55 < x < -1.2e39

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if -1.2e39 < x < -1.6500000000000001e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 78.1%

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

   if -1.6500000000000001e-18 < x < 3.2999999999999998e71

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.4%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+55}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+71}:\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+55)
   1.0
   (if (<= x -1e+39)
     -1.0
     (if (<= x -2e-18) 1.0 (if (<= x 7.5e+42) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+55) {
		tmp = 1.0;
	} else if (x <= -1e+39) {
		tmp = -1.0;
	} else if (x <= -2e-18) {
		tmp = 1.0;
	} else if (x <= 7.5e+42) {
		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 <= (-5d+55)) then
        tmp = 1.0d0
    else if (x <= (-1d+39)) then
        tmp = -1.0d0
    else if (x <= (-2d-18)) then
        tmp = 1.0d0
    else if (x <= 7.5d+42) 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 <= -5e+55) {
		tmp = 1.0;
	} else if (x <= -1e+39) {
		tmp = -1.0;
	} else if (x <= -2e-18) {
		tmp = 1.0;
	} else if (x <= 7.5e+42) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+55:
		tmp = 1.0
	elif x <= -1e+39:
		tmp = -1.0
	elif x <= -2e-18:
		tmp = 1.0
	elif x <= 7.5e+42:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+55)
		tmp = 1.0;
	elseif (x <= -1e+39)
		tmp = -1.0;
	elseif (x <= -2e-18)
		tmp = 1.0;
	elseif (x <= 7.5e+42)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+55)
		tmp = 1.0;
	elseif (x <= -1e+39)
		tmp = -1.0;
	elseif (x <= -2e-18)
		tmp = 1.0;
	elseif (x <= 7.5e+42)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+55], 1.0, If[LessEqual[x, -1e+39], -1.0, If[LessEqual[x, -2e-18], 1.0, If[LessEqual[x, 7.5e+42], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000046e55 or -9.9999999999999994e38 < x < -2.0000000000000001e-18 or 7.50000000000000041e42 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 81.6%

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

   if -5.00000000000000046e55 < x < -9.9999999999999994e38 or -2.0000000000000001e-18 < x < 7.50000000000000041e42

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 76.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 7: 50.1% 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 99.9%

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

3. Taylor expanded in x around 0 49.5%

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

4. Final simplification 49.5%

   \[\leadsto -1 \]
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 2024096 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

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