Kahan p9 Example

Percentage Accurate: 68.0% → 100.0%

Time: 5.4s

Alternatives: 7

Speedup: 2.1×

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: 68.0% 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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (/ (+ x y) (hypot x y)) (/ (hypot x y) (- x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

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

   1. add-sqr-sqrt 69.1%

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

   2. times-frac 69.9%

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

   3. hypot-def 69.9%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (* (/ (+ x y) (hypot x y)) (/ (- x y) (hypot x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
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 69.1%

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

   1. add-sqr-sqrt 69.1%

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

   2. times-frac 69.9%

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

   3. hypot-def 69.9%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-166}:\\ \;\;\;\;1 + \frac{-2}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e-166)
   (+ 1.0 (/ -2.0 (/ (/ x y) (/ y x))))
   (if (<= y 5e-69)
     (/ (* (+ x y) (- x y)) (+ (* x x) (* y y)))
     (+ (* 2.0 (* (/ x y) (/ x y))) -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-166) {
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	} else if (y <= 5e-69) {
		tmp = ((x + y) * (x - y)) / ((x * x) + (y * y));
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.5d-166) then
        tmp = 1.0d0 + ((-2.0d0) / ((x / y) / (y / x)))
    else if (y <= 5d-69) then
        tmp = ((x + y) * (x - y)) / ((x * x) + (y * y))
    else
        tmp = (2.0d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-166) {
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	} else if (y <= 5e-69) {
		tmp = ((x + y) * (x - y)) / ((x * x) + (y * y));
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 6.5e-166:
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)))
	elif y <= 5e-69:
		tmp = ((x + y) * (x - y)) / ((x * x) + (y * y))
	else:
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-166)
		tmp = Float64(1.0 + Float64(-2.0 / Float64(Float64(x / y) / Float64(y / x))));
	elseif (y <= 5e-69)
		tmp = Float64(Float64(Float64(x + y) * Float64(x - y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e-166)
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	elseif (y <= 5e-69)
		tmp = ((x + y) * (x - y)) / ((x * x) + (y * y));
	else
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 6.5e-166], N[(1.0 + N[(-2.0 / N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-69], N[(N[(N[(x + y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.50000000000000019e-166

   1. Initial program 62.6%

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

      1. add-sqr-sqrt 62.6%

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

      2. times-frac 63.5%

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

      3. hypot-def 63.6%

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

      4. hypot-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 27.5%

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

      1. associate-*r/ 27.5%

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

      2. associate-/l* 27.5%

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

      3. unpow2 27.5%

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

      4. unpow2 27.5%

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

      5. times-frac 39.4%

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

      6. unpow2 39.4%

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

   6. Simplified 39.4%

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

      1. pow2 39.4%

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

      2. clear-num 39.4%

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

      3. un-div-inv 39.4%

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

   8. Applied egg-rr 39.4%

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

   if 6.50000000000000019e-166 < y < 5.00000000000000033e-69

   1. Initial program 100.0%

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

   if 5.00000000000000033e-69 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-166}:\\ \;\;\;\;1 + \frac{-2}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]

Alternative 4: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-156} \lor \neg \left(y \leq 7.2 \cdot 10^{-146}\right) \land y \leq 5.6 \cdot 10^{-133}:\\ \;\;\;\;1 + \frac{-2}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.6e-156) (and (not (<= y 7.2e-146)) (<= y 5.6e-133)))
   (+ 1.0 (/ -2.0 (/ (/ x y) (/ y x))))
   -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((y <= 1.6e-156) || (!(y <= 7.2e-146) && (y <= 5.6e-133))) {
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 1.6d-156) .or. (.not. (y <= 7.2d-146)) .and. (y <= 5.6d-133)) then
        tmp = 1.0d0 + ((-2.0d0) / ((x / y) / (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((y <= 1.6e-156) || (!(y <= 7.2e-146) && (y <= 5.6e-133))) {
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if (y <= 1.6e-156) or (not (y <= 7.2e-146) and (y <= 5.6e-133)):
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if ((y <= 1.6e-156) || (!(y <= 7.2e-146) && (y <= 5.6e-133)))
		tmp = Float64(1.0 + Float64(-2.0 / Float64(Float64(x / y) / Float64(y / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 1.6e-156) || (~((y <= 7.2e-146)) && (y <= 5.6e-133)))
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[Or[LessEqual[y, 1.6e-156], And[N[Not[LessEqual[y, 7.2e-146]], $MachinePrecision], LessEqual[y, 5.6e-133]]], N[(1.0 + N[(-2.0 / N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.59999999999999991e-156 or 7.19999999999999957e-146 < y < 5.5999999999999997e-133

   1. Initial program 63.9%

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

      1. add-sqr-sqrt 63.9%

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

      2. times-frac 64.9%

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

      3. hypot-def 64.9%

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

      4. hypot-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 29.7%

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

      1. associate-*r/ 29.7%

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

      2. associate-/l* 29.7%

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

      3. unpow2 29.7%

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

      4. unpow2 29.7%

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

      5. times-frac 41.2%

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

      6. unpow2 41.2%

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

   6. Simplified 41.2%

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

      1. pow2 41.2%

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

      2. clear-num 41.2%

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

      3. un-div-inv 41.2%

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

   8. Applied egg-rr 41.2%

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

   if 1.59999999999999991e-156 < y < 7.19999999999999957e-146 or 5.5999999999999997e-133 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.3%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-156} \lor \neg \left(y \leq 7.2 \cdot 10^{-146}\right) \land y \leq 5.6 \cdot 10^{-133}:\\ \;\;\;\;1 + \frac{-2}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-163} \lor \neg \left(y \leq 1.48 \cdot 10^{-145}\right) \land y \leq 1.1 \cdot 10^{-130}:\\ \;\;\;\;1 + \frac{-2}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (or (<= y 6.5e-163) (and (not (<= y 1.48e-145)) (<= y 1.1e-130)))
   (+ 1.0 (/ -2.0 (/ (/ x y) (/ y x))))
   (+ (* 2.0 (* (/ x y) (/ x y))) -1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((y <= 6.5e-163) || (!(y <= 1.48e-145) && (y <= 1.1e-130))) {
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 6.5d-163) .or. (.not. (y <= 1.48d-145)) .and. (y <= 1.1d-130)) then
        tmp = 1.0d0 + ((-2.0d0) / ((x / y) / (y / x)))
    else
        tmp = (2.0d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((y <= 6.5e-163) || (!(y <= 1.48e-145) && (y <= 1.1e-130))) {
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if (y <= 6.5e-163) or (not (y <= 1.48e-145) and (y <= 1.1e-130)):
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)))
	else:
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if ((y <= 6.5e-163) || (!(y <= 1.48e-145) && (y <= 1.1e-130)))
		tmp = Float64(1.0 + Float64(-2.0 / Float64(Float64(x / y) / Float64(y / x))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 6.5e-163) || (~((y <= 1.48e-145)) && (y <= 1.1e-130)))
		tmp = 1.0 + (-2.0 / ((x / y) / (y / x)));
	else
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[Or[LessEqual[y, 6.5e-163], And[N[Not[LessEqual[y, 1.48e-145]], $MachinePrecision], LessEqual[y, 1.1e-130]]], N[(1.0 + N[(-2.0 / N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999999e-163 or 1.47999999999999995e-145 < y < 1.0999999999999999e-130

   1. Initial program 63.8%

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

      1. add-sqr-sqrt 63.8%

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

      2. times-frac 64.7%

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

      3. hypot-def 64.7%

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

      4. hypot-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 29.4%

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

      1. associate-*r/ 29.4%

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

      2. associate-/l* 29.4%

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

      3. unpow2 29.4%

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

      4. unpow2 29.4%

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

      5. times-frac 40.9%

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

      6. unpow2 40.9%

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

   6. Simplified 40.9%

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

      1. pow2 40.9%

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

      2. clear-num 40.9%

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

      3. un-div-inv 40.9%

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

   8. Applied egg-rr 40.9%

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

   if 6.4999999999999999e-163 < y < 1.47999999999999995e-145 or 1.0999999999999999e-130 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.4%

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

      1. unpow2 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 80.4%

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

   4. Applied egg-rr 80.4%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-163} \lor \neg \left(y \leq 1.48 \cdot 10^{-145}\right) \land y \leq 1.1 \cdot 10^{-130}:\\ \;\;\;\;1 + \frac{-2}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]

Alternative 6: 83.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-145}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 4.9e-157)
   1.0
   (if (<= y 1.06e-145) -1.0 (if (<= y 3.8e-132) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-157) {
		tmp = 1.0;
	} else if (y <= 1.06e-145) {
		tmp = -1.0;
	} else if (y <= 3.8e-132) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.9d-157) then
        tmp = 1.0d0
    else if (y <= 1.06d-145) then
        tmp = -1.0d0
    else if (y <= 3.8d-132) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-157) {
		tmp = 1.0;
	} else if (y <= 1.06e-145) {
		tmp = -1.0;
	} else if (y <= 3.8e-132) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 4.9e-157:
		tmp = 1.0
	elif y <= 1.06e-145:
		tmp = -1.0
	elif y <= 3.8e-132:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 4.9e-157)
		tmp = 1.0;
	elseif (y <= 1.06e-145)
		tmp = -1.0;
	elseif (y <= 3.8e-132)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.9e-157)
		tmp = 1.0;
	elseif (y <= 1.06e-145)
		tmp = -1.0;
	elseif (y <= 3.8e-132)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 4.9e-157], 1.0, If[LessEqual[y, 1.06e-145], -1.0, If[LessEqual[y, 3.8e-132], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.8999999999999998e-157 or 1.05999999999999996e-145 < y < 3.7999999999999997e-132

   1. Initial program 63.9%

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

   2. Taylor expanded in x around inf 39.3%

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

   if 4.8999999999999998e-157 < y < 1.05999999999999996e-145 or 3.7999999999999997e-132 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.3%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-145}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 66.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ -1 \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	return -1.0;
}

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

y = abs(y)
function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 69.1%

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

2. Taylor expanded in x around 0 64.4%

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

3. Final simplification 64.4%

   \[\leadsto -1 \]

Developer target: 100.0% 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 2023299 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (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))))