Kahan p9 Example

Percentage Accurate: 68.6% → 99.9%

Time: 10.7s

Alternatives: 11

Speedup: 0.4×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x, y):
	return ((x - y) / math.hypot(x, y)) * ((x + y) / math.hypot(x, y))

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.6%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 67.6%

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

   2. times-frac 67.9%

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

   3. hypot-define 67.9%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 72.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (* (/ (+ x y) (hypot x y)) (+ (/ x y) -1.0)))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((x + y) / hypot(x, y)) * ((x / y) + -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((x + y) / Math.hypot(x, y)) * ((x / y) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = ((x + y) / math.hypot(x, y)) * ((x / y) + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x + y) / hypot(x, y)) * Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = ((x + y) / hypot(x, y)) * ((x / y) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-define 3.1%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 12.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (+ (* 2.0 (pow (/ x y) 2.0)) -1.0))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * pow((x / y), 2.0)) + -1.0;
	}
	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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (2.0d0 * ((x / y) ** 2.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * Math.pow((x / y), 2.0)) + -1.0;
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * (Float64(x / y) ^ 2.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = (2.0 * ((x / y) ^ 2.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-define 3.1%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div-sub 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 55.4%

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

      1. fma-neg 55.4%

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

      2. unpow2 55.4%

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

      3. unpow2 55.4%

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

      4. times-frac 74.9%

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

      5. unpow2 74.9%

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

      6. metadata-eval 74.9%

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

   9. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]
   10. Step-by-step derivation

       1. fma-undefine 74.9%

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

   11. Applied egg-rr 74.9%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{x}{y}\right)}^{2} + -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 41.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + 2 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ t_1 := \frac{x - y}{x + y \cdot \left(-1 + \frac{y}{x}\right)}\\ \mathbf{if}\;y \leq 1.3 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{\left(x - y\right) + \left(x - y\right) \cdot \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* 2.0 (/ x (* y (/ y x))))))
        (t_1 (/ (- x y) (+ x (* y (+ -1.0 (/ y x)))))))
   (if (<= y 1.3e-169)
     t_1
     (if (<= y 2.7e-101)
       t_0
       (if (<= y 6.8e-99)
         t_1
         (if (<= y 5.8e-27)
           (/ (+ (- x y) (* (- x y) (/ x y))) y)
           (if (<= y 6e-27) 1.0 t_0)))))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (2.0 * (x / (y * (y / x))));
	double t_1 = (x - y) / (x + (y * (-1.0 + (y / x))));
	double tmp;
	if (y <= 1.3e-169) {
		tmp = t_1;
	} else if (y <= 2.7e-101) {
		tmp = t_0;
	} else if (y <= 6.8e-99) {
		tmp = t_1;
	} else if (y <= 5.8e-27) {
		tmp = ((x - y) + ((x - y) * (x / y))) / y;
	} else if (y <= 6e-27) {
		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) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + (2.0d0 * (x / (y * (y / x))))
    t_1 = (x - y) / (x + (y * ((-1.0d0) + (y / x))))
    if (y <= 1.3d-169) then
        tmp = t_1
    else if (y <= 2.7d-101) then
        tmp = t_0
    else if (y <= 6.8d-99) then
        tmp = t_1
    else if (y <= 5.8d-27) then
        tmp = ((x - y) + ((x - y) * (x / y))) / y
    else if (y <= 6d-27) 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 * (x / (y * (y / x))));
	double t_1 = (x - y) / (x + (y * (-1.0 + (y / x))));
	double tmp;
	if (y <= 1.3e-169) {
		tmp = t_1;
	} else if (y <= 2.7e-101) {
		tmp = t_0;
	} else if (y <= 6.8e-99) {
		tmp = t_1;
	} else if (y <= 5.8e-27) {
		tmp = ((x - y) + ((x - y) * (x / y))) / y;
	} else if (y <= 6e-27) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (2.0 * (x / (y * (y / x))))
	t_1 = (x - y) / (x + (y * (-1.0 + (y / x))))
	tmp = 0
	if y <= 1.3e-169:
		tmp = t_1
	elif y <= 2.7e-101:
		tmp = t_0
	elif y <= 6.8e-99:
		tmp = t_1
	elif y <= 5.8e-27:
		tmp = ((x - y) + ((x - y) * (x / y))) / y
	elif y <= 6e-27:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(2.0 * Float64(x / Float64(y * Float64(y / x)))))
	t_1 = Float64(Float64(x - y) / Float64(x + Float64(y * Float64(-1.0 + Float64(y / x)))))
	tmp = 0.0
	if (y <= 1.3e-169)
		tmp = t_1;
	elseif (y <= 2.7e-101)
		tmp = t_0;
	elseif (y <= 6.8e-99)
		tmp = t_1;
	elseif (y <= 5.8e-27)
		tmp = Float64(Float64(Float64(x - y) + Float64(Float64(x - y) * Float64(x / y))) / y);
	elseif (y <= 6e-27)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + (2.0 * (x / (y * (y / x))));
	t_1 = (x - y) / (x + (y * (-1.0 + (y / x))));
	tmp = 0.0;
	if (y <= 1.3e-169)
		tmp = t_1;
	elseif (y <= 2.7e-101)
		tmp = t_0;
	elseif (y <= 6.8e-99)
		tmp = t_1;
	elseif (y <= 5.8e-27)
		tmp = ((x - y) + ((x - y) * (x / y))) / y;
	elseif (y <= 6e-27)
		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[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(x + N[(y * N[(-1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.3e-169], t$95$1, If[LessEqual[y, 2.7e-101], t$95$0, If[LessEqual[y, 6.8e-99], t$95$1, If[LessEqual[y, 5.8e-27], N[(N[(N[(x - y), $MachinePrecision] + N[(N[(x - y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 6e-27], 1.0, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.30000000000000007e-169 or 2.7000000000000002e-101 < y < 6.80000000000000014e-99

   1. Initial program 61.9%

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

      1. associate-/l* 62.4%

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

      2. +-commutative 62.4%

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

      3. fma-define 62.4%

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

   3. Simplified 62.4%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 37.6%

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

      1. clear-num 37.6%

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

      2. un-div-inv 37.7%

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

   7. Applied egg-rr 37.7%

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

   8. Taylor expanded in y around 0 37.0%

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

   if 1.30000000000000007e-169 < y < 2.7000000000000002e-101 or 6.0000000000000002e-27 < y

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. times-frac 96.0%

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

      3. hypot-define 96.2%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div-sub 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 76.9%

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

      1. fma-neg 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

      4. times-frac 77.2%

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

      5. unpow2 77.2%

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

      6. metadata-eval 77.2%

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

   9. Simplified 77.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]
   10. Step-by-step derivation

       1. fma-undefine 77.2%

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

   11. Applied egg-rr 77.2%

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

       1. unpow2 77.2%

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

       2. clear-num 77.2%

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

       3. frac-times 77.2%

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

       4. *-un-lft-identity 77.2%

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

   13. Applied egg-rr 77.2%

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

   if 6.80000000000000014e-99 < y < 5.80000000000000008e-27

   1. Initial program 100.0%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 92.1%

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

      1. associate-*r/ 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. distribute-rgt-in 92.1%

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

      2. *-un-lft-identity 92.1%

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

   9. Applied egg-rr 92.1%

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

   if 5.80000000000000008e-27 < y < 6.0000000000000002e-27

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-169}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(-1 + \frac{y}{x}\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-101}:\\ \;\;\;\;-1 + 2 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(-1 + \frac{y}{x}\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{\left(x - y\right) + \left(x - y\right) \cdot \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -1.0 + (2.0 * (x / (y * (y / x))));
	}
	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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (-1.0d0) + (2.0d0 * (x / (y * (y / x))))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = -1.0 + (2.0 * (x / (y * (y / x))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(-1.0 + Float64(2.0 * Float64(x / Float64(y * Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = -1.0 + (2.0 * (x / (y * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-define 3.1%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div-sub 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 55.4%

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

      1. fma-neg 55.4%

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

      2. unpow2 55.4%

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

      3. unpow2 55.4%

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

      4. times-frac 74.9%

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

      5. unpow2 74.9%

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

      6. metadata-eval 74.9%

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

   9. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]
   10. Step-by-step derivation

       1. fma-undefine 74.9%

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

   11. Applied egg-rr 74.9%

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

       1. unpow2 74.9%

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

       2. clear-num 74.9%

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

       3. frac-times 74.7%

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

       4. *-un-lft-identity 74.7%

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

   13. Applied egg-rr 74.7%

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

4. Final simplification 91.8%

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

Alternative 6: 40.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e-160)
   1.0
   (if (<= y 7.5e-123)
     -1.0
     (if (<= y 1.15e-122)
       1.0
       (if (<= y 9.5e-45) -1.0 (if (<= y 9.8e-45) 1.0 -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-160) {
		tmp = 1.0;
	} else if (y <= 7.5e-123) {
		tmp = -1.0;
	} else if (y <= 1.15e-122) {
		tmp = 1.0;
	} else if (y <= 9.5e-45) {
		tmp = -1.0;
	} else if (y <= 9.8e-45) {
		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 <= 4d-160) then
        tmp = 1.0d0
    else if (y <= 7.5d-123) then
        tmp = -1.0d0
    else if (y <= 1.15d-122) then
        tmp = 1.0d0
    else if (y <= 9.5d-45) then
        tmp = -1.0d0
    else if (y <= 9.8d-45) 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 <= 4e-160) {
		tmp = 1.0;
	} else if (y <= 7.5e-123) {
		tmp = -1.0;
	} else if (y <= 1.15e-122) {
		tmp = 1.0;
	} else if (y <= 9.5e-45) {
		tmp = -1.0;
	} else if (y <= 9.8e-45) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-160:
		tmp = 1.0
	elif y <= 7.5e-123:
		tmp = -1.0
	elif y <= 1.15e-122:
		tmp = 1.0
	elif y <= 9.5e-45:
		tmp = -1.0
	elif y <= 9.8e-45:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-160)
		tmp = 1.0;
	elseif (y <= 7.5e-123)
		tmp = -1.0;
	elseif (y <= 1.15e-122)
		tmp = 1.0;
	elseif (y <= 9.5e-45)
		tmp = -1.0;
	elseif (y <= 9.8e-45)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-160)
		tmp = 1.0;
	elseif (y <= 7.5e-123)
		tmp = -1.0;
	elseif (y <= 1.15e-122)
		tmp = 1.0;
	elseif (y <= 9.5e-45)
		tmp = -1.0;
	elseif (y <= 9.8e-45)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-160], 1.0, If[LessEqual[y, 7.5e-123], -1.0, If[LessEqual[y, 1.15e-122], 1.0, If[LessEqual[y, 9.5e-45], -1.0, If[LessEqual[y, 9.8e-45], 1.0, -1.0]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 4e-160 or 7.50000000000000011e-123 < y < 1.15000000000000003e-122 or 9.5000000000000002e-45 < y < 9.7999999999999996e-45

   1. Initial program 61.8%

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

      1. associate-/l* 62.1%

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

      2. +-commutative 62.1%

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

      3. fma-define 62.1%

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

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 36.2%

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

   if 4e-160 < y < 7.50000000000000011e-123 or 1.15000000000000003e-122 < y < 9.5000000000000002e-45 or 9.7999999999999996e-45 < y

   1. Initial program 100.0%

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

      1. associate-/l* 98.0%

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

      2. +-commutative 98.0%

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

      3. fma-define 98.0%

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

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 79.8%

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

Alternative 7: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;1 + -1.5 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-197)
   (+ 1.0 (* -1.5 (/ (/ y x) (/ x y))))
   (+ -1.0 (* 2.0 (/ x (* y (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-197) {
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0 + (2.0 * (x / (y * (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 <= 1.3d-197) then
        tmp = 1.0d0 + ((-1.5d0) * ((y / x) / (x / y)))
    else
        tmp = (-1.0d0) + (2.0d0 * (x / (y * (y / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-197) {
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0 + (2.0 * (x / (y * (y / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-197:
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)))
	else:
		tmp = -1.0 + (2.0 * (x / (y * (y / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-197)
		tmp = Float64(1.0 + Float64(-1.5 * Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = Float64(-1.0 + Float64(2.0 * Float64(x / Float64(y * Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-197)
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	else
		tmp = -1.0 + (2.0 * (x / (y * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-197], N[(1.0 + N[(-1.5 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 * N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-197

   1. Initial program 62.1%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 62.1%

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

      2. times-frac 62.9%

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

      3. hypot-define 62.9%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 37.9%

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

   6. Taylor expanded in y around 0 27.2%

      \[\leadsto \color{blue}{1 + -1.5 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. +-commutative 27.2%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{{y}^{2}}{{x}^{2}} + 1} \]

      2. fma-define 27.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1.5, \frac{{y}^{2}}{{x}^{2}}, 1\right)} \]

      3. unpow2 27.2%

         \[\leadsto \mathsf{fma}\left(-1.5, \frac{\color{blue}{y \cdot y}}{{x}^{2}}, 1\right) \]

      4. unpow2 27.2%

         \[\leadsto \mathsf{fma}\left(-1.5, \frac{y \cdot y}{\color{blue}{x \cdot x}}, 1\right) \]

      5. times-frac 37.4%

         \[\leadsto \mathsf{fma}\left(-1.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right) \]

      6. unpow2 37.4%

         \[\leadsto \mathsf{fma}\left(-1.5, \color{blue}{{\left(\frac{y}{x}\right)}^{2}}, 1\right) \]

   8. Simplified 37.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 37.4%

         \[\leadsto \color{blue}{-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + 1} \]

   10. Applied egg-rr 37.4%

       \[\leadsto \color{blue}{-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + 1} \]
   11. Step-by-step derivation

       1. unpow2 37.4%

          \[\leadsto -1.5 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} + 1 \]

       2. clear-num 37.4%

          \[\leadsto -1.5 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) + 1 \]

       3. un-div-inv 37.4%

          \[\leadsto -1.5 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} + 1 \]

   12. Applied egg-rr 37.4%

       \[\leadsto -1.5 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} + 1 \]

   if 1.3000000000000001e-197 < y

   1. Initial program 93.3%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 93.3%

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

      2. times-frac 91.3%

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

      3. hypot-define 91.4%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div-sub 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 69.9%

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

      1. fma-neg 69.9%

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

      2. unpow2 69.9%

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

      3. unpow2 69.9%

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

      4. times-frac 76.9%

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

      5. unpow2 76.9%

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

      6. metadata-eval 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]
   10. Step-by-step derivation

       1. fma-undefine 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. unpow2 76.9%

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

       2. clear-num 76.9%

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

       3. frac-times 76.9%

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

       4. *-un-lft-identity 76.9%

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

   13. Applied egg-rr 76.9%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;1 + -1.5 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;1 + -1.5 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-197)
   (+ 1.0 (* -1.5 (/ (/ y x) (/ x y))))
   (* (- x y) (/ (+ (/ x y) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-197) {
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-197) then
        tmp = 1.0d0 + ((-1.5d0) * ((y / x) / (x / y)))
    else
        tmp = (x - y) * (((x / y) + 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-197) {
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-197:
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)))
	else:
		tmp = (x - y) * (((x / y) + 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-197)
		tmp = Float64(1.0 + Float64(-1.5 * Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-197)
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	else
		tmp = (x - y) * (((x / y) + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-197], N[(1.0 + N[(-1.5 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-197

   1. Initial program 62.1%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 62.1%

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

      2. times-frac 62.9%

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

      3. hypot-define 62.9%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 37.9%

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

   6. Taylor expanded in y around 0 27.2%

      \[\leadsto \color{blue}{1 + -1.5 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. +-commutative 27.2%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{{y}^{2}}{{x}^{2}} + 1} \]

      2. fma-define 27.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1.5, \frac{{y}^{2}}{{x}^{2}}, 1\right)} \]

      3. unpow2 27.2%

         \[\leadsto \mathsf{fma}\left(-1.5, \frac{\color{blue}{y \cdot y}}{{x}^{2}}, 1\right) \]

      4. unpow2 27.2%

         \[\leadsto \mathsf{fma}\left(-1.5, \frac{y \cdot y}{\color{blue}{x \cdot x}}, 1\right) \]

      5. times-frac 37.4%

         \[\leadsto \mathsf{fma}\left(-1.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right) \]

      6. unpow2 37.4%

         \[\leadsto \mathsf{fma}\left(-1.5, \color{blue}{{\left(\frac{y}{x}\right)}^{2}}, 1\right) \]

   8. Simplified 37.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 37.4%

         \[\leadsto \color{blue}{-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + 1} \]

   10. Applied egg-rr 37.4%

       \[\leadsto \color{blue}{-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + 1} \]
   11. Step-by-step derivation

       1. unpow2 37.4%

          \[\leadsto -1.5 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} + 1 \]

       2. clear-num 37.4%

          \[\leadsto -1.5 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) + 1 \]

       3. un-div-inv 37.4%

          \[\leadsto -1.5 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} + 1 \]

   12. Applied egg-rr 37.4%

       \[\leadsto -1.5 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} + 1 \]

   if 1.3000000000000001e-197 < y

   1. Initial program 93.3%

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

      1. associate-/l* 91.0%

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

      2. +-commutative 91.0%

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

      3. fma-define 91.0%

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

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 76.7%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;1 + -1.5 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;\left(1 + \frac{y}{x}\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-197)
   (* (+ 1.0 (/ y x)) (- 1.0 (/ y x)))
   (* (- x y) (/ (+ (/ x y) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-197) {
		tmp = (1.0 + (y / x)) * (1.0 - (y / x));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-197) then
        tmp = (1.0d0 + (y / x)) * (1.0d0 - (y / x))
    else
        tmp = (x - y) * (((x / y) + 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-197) {
		tmp = (1.0 + (y / x)) * (1.0 - (y / x));
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-197:
		tmp = (1.0 + (y / x)) * (1.0 - (y / x))
	else:
		tmp = (x - y) * (((x / y) + 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-197)
		tmp = Float64(Float64(1.0 + Float64(y / x)) * Float64(1.0 - Float64(y / x)));
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-197)
		tmp = (1.0 + (y / x)) * (1.0 - (y / x));
	else
		tmp = (x - y) * (((x / y) + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-197], N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-197

   1. Initial program 62.1%

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

      1. associate-/l* 62.6%

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

      2. +-commutative 62.6%

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

      3. fma-define 62.6%

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

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 37.3%

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

      1. clear-num 37.3%

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

      2. un-div-inv 37.4%

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

   7. Applied egg-rr 37.4%

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

      1. div-inv 37.3%

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

      2. clear-num 37.3%

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

   9. Applied egg-rr 37.3%

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

       1. associate-*r/ 37.4%

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

       2. associate-*l/ 37.4%

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

       3. *-commutative 37.4%

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

       4. div-sub 37.4%

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

       5. *-inverses 37.4%

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

   11. Simplified 37.4%

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

   if 1.3000000000000001e-197 < y

   1. Initial program 93.3%

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

      1. associate-/l* 91.0%

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

      2. +-commutative 91.0%

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

      3. fma-define 91.0%

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

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 76.7%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;\left(1 + \frac{y}{x}\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-181}:\\ \;\;\;\;\left(1 + \frac{y}{x}\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-181) (* (+ 1.0 (/ y x)) (- 1.0 (/ y x))) -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-181) {
		tmp = (1.0 + (y / x)) * (1.0 - (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 (y <= 2.2d-181) then
        tmp = (1.0d0 + (y / x)) * (1.0d0 - (y / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-181) {
		tmp = (1.0 + (y / x)) * (1.0 - (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-181:
		tmp = (1.0 + (y / x)) * (1.0 - (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-181)
		tmp = Float64(Float64(1.0 + Float64(y / x)) * Float64(1.0 - Float64(y / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-181)
		tmp = (1.0 + (y / x)) * (1.0 - (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-181], N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.19999999999999997e-181

   1. Initial program 61.9%

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

      1. associate-/l* 62.4%

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

      2. +-commutative 62.4%

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

      3. fma-define 62.4%

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

   3. Simplified 62.4%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 37.6%

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

      1. clear-num 37.6%

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

      2. un-div-inv 37.7%

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

   7. Applied egg-rr 37.7%

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

      1. div-inv 37.6%

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

      2. clear-num 37.6%

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

   9. Applied egg-rr 37.6%

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

       1. associate-*r/ 37.7%

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

       2. associate-*l/ 37.7%

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

       3. *-commutative 37.7%

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

       4. div-sub 37.7%

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

       5. *-inverses 37.7%

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

   11. Simplified 37.7%

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

   if 2.19999999999999997e-181 < y

   1. Initial program 97.6%

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

      1. associate-/l* 94.9%

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

      2. +-commutative 94.9%

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

      3. fma-define 94.9%

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

   3. Simplified 94.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 78.4%

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

Alternative 11: 66.8% accurate, 15.0× 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 67.6%

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

   1. associate-/l* 67.6%

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

   2. +-commutative 67.6%

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

   3. fma-define 67.6%

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

3. Simplified 67.6%

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 66.7%

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

Developer target: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t\_0 \land t\_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (and (< 0.5 t_0) (< t_0 2.0))
     (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))
     (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = fabs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	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 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(2.0 / Float64(1.0 + Float64(Float64(x / y) * Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[And[Less[0.5, t$95$0], Less[t$95$0, 2.0]], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024107 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

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