Kahan p9 Example

Percentage Accurate: 67.4% → 99.9%

Time: 9.7s

Alternatives: 10

Speedup: 15.0×

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: 67.4% 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 64.4%

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

   1. add-sqr-sqrt 64.4%

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

   2. times-frac 65.2%

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

   3. hypot-define 65.2%

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

   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. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. associate-/l* 64.9%

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

   2. +-commutative 64.9%

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

   3. fma-define 64.9%

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

3. Simplified 64.9%

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

   1. fma-undefine 64.9%

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

   2. +-commutative 64.9%

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

   3. *-un-lft-identity 64.9%

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

   4. add-sqr-sqrt 64.9%

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

   5. times-frac 65.0%

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

   6. hypot-define 65.1%

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

   7. hypot-define 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.8%

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

   2. *-un-lft-identity 99.8%

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

8. Applied egg-rr 99.8%

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

Alternative 3: 46.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;1 - \frac{\left(y \cdot 2\right) \cdot \frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e-162)
   (- 1.0 (/ (* (* y 2.0) (/ y x)) x))
   (if (<= y 4e-35) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-162) {
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	} else if (y <= 4e-35) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.6d-162) then
        tmp = 1.0d0 - (((y * 2.0d0) * (y / x)) / x)
    else if (y <= 4d-35) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-162) {
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	} else if (y <= 4e-35) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.6e-162:
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x)
	elif y <= 4e-35:
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-162)
		tmp = Float64(1.0 - Float64(Float64(Float64(y * 2.0) * Float64(y / x)) / x));
	elseif (y <= 4e-35)
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e-162)
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	elseif (y <= 4e-35)
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e-162], N[(1.0 - N[(N[(N[(y * 2.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e-35], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.59999999999999988e-162

   1. Initial program 58.4%

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

      1. associate-/l* 59.0%

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

      2. +-commutative 59.0%

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

      3. fma-define 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in x around -inf 34.6%

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

   7. Simplified 35.4%

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

      1. unpow2 35.4%

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

   9. Applied egg-rr 35.4%

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

       1. associate-*r* 35.4%

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

       2. *-un-lft-identity 35.4%

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

       3. times-frac 35.9%

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

   11. Applied egg-rr 35.9%

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

       1. /-rgt-identity 35.9%

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

   13. Simplified 35.9%

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

   if 1.59999999999999988e-162 < y < 4.00000000000000003e-35

   1. Initial program 99.9%

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

   if 4.00000000000000003e-35 < 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* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;1 - \frac{\left(y \cdot 2\right) \cdot \frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{\left(y \cdot 2\right) \cdot \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e-145)
   (- 1.0 (/ (* (* y 2.0) (/ y x)) x))
   (/ (* (- x y) (+ 1.0 (/ x y))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-145) {
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	} else {
		tmp = ((x - y) * (1.0 + (x / y))) / 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 <= 6.4d-145) then
        tmp = 1.0d0 - (((y * 2.0d0) * (y / x)) / x)
    else
        tmp = ((x - y) * (1.0d0 + (x / y))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-145) {
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	} else {
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.4e-145:
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x)
	else:
		tmp = ((x - y) * (1.0 + (x / y))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.4e-145)
		tmp = Float64(1.0 - Float64(Float64(Float64(y * 2.0) * Float64(y / x)) / x));
	else
		tmp = Float64(Float64(Float64(x - y) * Float64(1.0 + Float64(x / y))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e-145)
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	else
		tmp = ((x - y) * (1.0 + (x / y))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.4e-145], N[(1.0 - N[(N[(N[(y * 2.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.40000000000000017e-145

   1. Initial program 58.6%

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

      1. associate-/l* 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-define 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in x around -inf 34.9%

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

   7. Simplified 35.7%

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

      1. unpow2 35.7%

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

   9. Applied egg-rr 35.7%

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

       1. associate-*r* 35.7%

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

       2. *-un-lft-identity 35.7%

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

       3. times-frac 36.2%

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

   11. Applied egg-rr 36.2%

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

       1. /-rgt-identity 36.2%

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

   13. Simplified 36.2%

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

   if 6.40000000000000017e-145 < 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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 83.3%

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

      1. associate-*r/ 83.4%

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

   7. Applied egg-rr 83.4%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{\left(y \cdot 2\right) \cdot \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-143}:\\ \;\;\;\;1 - \frac{\left(y \cdot 2\right) \cdot \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-143)
   (- 1.0 (/ (* (* y 2.0) (/ y x)) x))
   (* (+ 1.0 (/ x y)) (+ -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-143) {
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / 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 <= 2.2d-143) then
        tmp = 1.0d0 - (((y * 2.0d0) * (y / x)) / x)
    else
        tmp = (1.0d0 + (x / y)) * ((-1.0d0) + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-143) {
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-143:
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x)
	else:
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-143)
		tmp = Float64(1.0 - Float64(Float64(Float64(y * 2.0) * Float64(y / x)) / x));
	else
		tmp = Float64(Float64(1.0 + Float64(x / y)) * Float64(-1.0 + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-143)
		tmp = 1.0 - (((y * 2.0) * (y / x)) / x);
	else
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-143], N[(1.0 - N[(N[(N[(y * 2.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.19999999999999989e-143

   1. Initial program 58.6%

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

      1. associate-/l* 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-define 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in x around -inf 34.9%

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

   7. Simplified 35.7%

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

      1. unpow2 35.7%

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

   9. Applied egg-rr 35.7%

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

       1. associate-*r* 35.7%

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

       2. *-un-lft-identity 35.7%

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

       3. times-frac 36.2%

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

   11. Applied egg-rr 36.2%

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

       1. /-rgt-identity 36.2%

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

   13. Simplified 36.2%

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

   if 2.19999999999999989e-143 < 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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 83.3%

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

      1. associate-*r/ 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-/l* 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. associate-*r/ 83.4%

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

       2. *-commutative 83.4%

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

       3. associate-*r/ 83.4%

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

       4. div-sub 83.4%

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

       5. sub-neg 83.4%

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

       6. *-inverses 83.4%

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

       7. metadata-eval 83.4%

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

   11. Simplified 83.4%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-143}:\\ \;\;\;\;1 - \frac{\left(y \cdot 2\right) \cdot \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{x - y}{x} \cdot \frac{x + y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e-145)
   (* (/ (- x y) x) (/ (+ x y) x))
   (* (+ 1.0 (/ x y)) (+ -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-145) {
		tmp = ((x - y) / x) * ((x + y) / x);
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / 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 <= 6.5d-145) then
        tmp = ((x - y) / x) * ((x + y) / x)
    else
        tmp = (1.0d0 + (x / y)) * ((-1.0d0) + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-145) {
		tmp = ((x - y) / x) * ((x + y) / x);
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.5e-145:
		tmp = ((x - y) / x) * ((x + y) / x)
	else:
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-145)
		tmp = Float64(Float64(Float64(x - y) / x) * Float64(Float64(x + y) / x));
	else
		tmp = Float64(Float64(1.0 + Float64(x / y)) * Float64(-1.0 + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e-145)
		tmp = ((x - y) / x) * ((x + y) / x);
	else
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.5e-145], N[(N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5000000000000002e-145

   1. Initial program 58.6%

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

      1. add-sqr-sqrt 58.6%

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

      2. times-frac 59.5%

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

      3. hypot-define 59.5%

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

      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 36.5%

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

   6. Taylor expanded in x around inf 36.0%

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

   if 6.5000000000000002e-145 < 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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 83.3%

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

      1. associate-*r/ 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-/l* 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. associate-*r/ 83.4%

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

       2. *-commutative 83.4%

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

       3. associate-*r/ 83.4%

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

       4. div-sub 83.4%

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

       5. sub-neg 83.4%

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

       6. *-inverses 83.4%

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

       7. metadata-eval 83.4%

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

   11. Simplified 83.4%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{x - y}{x} \cdot \frac{x + y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-145)
   (* (- x y) (/ (+ 1.0 (/ y x)) x))
   (* (+ 1.0 (/ x y)) (+ -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-145) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / 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 <= 8.5d-145) then
        tmp = (x - y) * ((1.0d0 + (y / x)) / x)
    else
        tmp = (1.0d0 + (x / y)) * ((-1.0d0) + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-145) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-145:
		tmp = (x - y) * ((1.0 + (y / x)) / x)
	else:
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-145)
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x));
	else
		tmp = Float64(Float64(1.0 + Float64(x / y)) * Float64(-1.0 + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-145)
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	else
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-145], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000043e-145

   1. Initial program 58.6%

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

      1. associate-/l* 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-define 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in x around inf 35.9%

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

   if 8.50000000000000043e-145 < 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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 83.3%

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

      1. associate-*r/ 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-/l* 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. associate-*r/ 83.4%

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

       2. *-commutative 83.4%

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

       3. associate-*r/ 83.4%

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

       4. div-sub 83.4%

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

       5. sub-neg 83.4%

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

       6. *-inverses 83.4%

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

       7. metadata-eval 83.4%

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

   11. Simplified 83.4%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e-143) 1.0 (* (+ 1.0 (/ x y)) (+ -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-143) {
		tmp = 1.0;
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / 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 <= 1.35d-143) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 + (x / y)) * ((-1.0d0) + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-143) {
		tmp = 1.0;
	} else {
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.35e-143:
		tmp = 1.0
	else:
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.35e-143)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 + Float64(x / y)) * Float64(-1.0 + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e-143)
		tmp = 1.0;
	else
		tmp = (1.0 + (x / y)) * (-1.0 + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.35e-143], 1.0, N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000005e-143

   1. Initial program 58.6%

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

      1. associate-/l* 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-define 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in x around inf 34.5%

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

   if 1.35000000000000005e-143 < 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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 83.3%

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

      1. associate-*r/ 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-/l* 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. associate-*r/ 83.4%

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

       2. *-commutative 83.4%

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

       3. associate-*r/ 83.4%

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

       4. div-sub 83.4%

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

       5. sub-neg 83.4%

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

       6. *-inverses 83.4%

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

       7. metadata-eval 83.4%

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

   11. Simplified 83.4%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.45e-144) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-144) {
		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 <= 1.45d-144) 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 <= 1.45e-144) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e-144:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-144)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.45e-144)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e-144], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.4500000000000001e-144

   1. Initial program 58.6%

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

      1. associate-/l* 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-define 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in x around inf 34.5%

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

   if 1.4500000000000001e-144 < 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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 82.3%

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

Alternative 10: 67.5% 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 64.4%

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

   1. associate-/l* 64.9%

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

   2. +-commutative 64.9%

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

   3. fma-define 64.9%

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

3. Simplified 64.9%

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

5. Taylor expanded in x around 0 68.7%

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

Developer Target 1: 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 2024131 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (! :herbie-platform default (if (< 1/2 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y)))))))

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