Kahan p9 Example

Percentage Accurate: 69.1% → 100.0%

Time: 9.4s

Alternatives: 8

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: 69.1% 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, 0.8× speedup

Math

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

FPCore

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

C

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

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 = (x - y) / ((x * (x / (x + y))) + (y * (y / (x + y))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. +-commutative 67.2%

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

   2. +-commutative 67.2%

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

   3. associate-*r/ 67.1%

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

   4. +-commutative 67.1%

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

   5. +-commutative 67.1%

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

   6. fma-def 67.1%

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

3. Simplified 67.1%

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

   1. associate-*r/ 67.2%

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

   2. associate-/l* 67.3%

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

5. Applied egg-rr 67.3%

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

   1. div-inv 67.0%

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

   2. *-commutative 67.0%

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

   3. fma-udef 67.0%

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

   4. distribute-rgt-in 67.0%

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

   5. div-inv 67.1%

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

   6. div-inv 67.3%

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

   7. flip-+ 66.9%

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

   8. flip-- 66.2%

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

   9. associate-/r/ 66.2%

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

   10. times-frac 66.9%

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

   11. *-inverses 90.9%

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

   12. /-rgt-identity 90.9%

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

7. Applied egg-rr 90.9%

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

   1. associate-/l* 100.0%

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

   2. associate-/r/ 100.0%

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

   3. +-commutative 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 92.7% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(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) y))))

C

y = abs(y);
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) / y;
	}
	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) :: 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 = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
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) / y;
	}
	return tmp;
}

Python

y = abs(y)
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) / y
	return tmp

Julia

y = abs(y)
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(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
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) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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[(x - y), $MachinePrecision] / y), $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%

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

   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. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. associate-*r/ 3.1%

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

      4. +-commutative 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 71.1%

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

   5. Taylor expanded in x around 0 71.2%

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

      1. *-inverses 71.2%

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

      2. div-sub 71.2%

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

   7. Simplified 71.2%

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

4. Final simplification 90.5%

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

Alternative 3: 93.1% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(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) (+ (- y x) (* 2.0 (* x (/ x y))))))))

C

y = abs(y);
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) / ((y - x) + (2.0 * (x * (x / y))));
	}
	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) :: 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 = (x - y) / ((y - x) + (2.0d0 * (x * (x / y))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
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) / ((y - x) + (2.0 * (x * (x / y))));
	}
	return tmp;
}

Python

y = abs(y)
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) / ((y - x) + (2.0 * (x * (x / y))))
	return tmp

Julia

y = abs(y)
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(x - y) / Float64(Float64(y - x) + Float64(2.0 * Float64(x * Float64(x / y)))));
	end
	return tmp
end

MATLAB

y = abs(y)
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) / ((y - x) + (2.0 * (x * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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[(x - y), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] + N[(2.0 * N[(x * N[(x / y), $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%

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

   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. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. associate-*r/ 3.1%

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

      4. +-commutative 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

   3. Simplified 3.1%

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

      1. associate-*r/ 0.0%

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

      2. associate-/l* 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 71.6%

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

      1. mul-1-neg 71.6%

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

      2. associate-+r+ 71.6%

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

      3. sub-neg 71.6%

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

      4. unpow2 71.6%

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

      5. associate-*r/ 72.6%

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

   8. Simplified 72.6%

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

4. Final simplification 91.0%

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

Alternative 4: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-138} \lor \neg \left(y \leq 1.75 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-190)
   1.0
   (if (or (<= y 2e-138) (not (<= y 1.75e-122)))
     (/ (- x y) y)
     (+ 1.0 (* -2.0 (/ (* y y) (* x x)))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-190) {
		tmp = 1.0;
	} else if ((y <= 2e-138) || !(y <= 1.75e-122)) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	}
	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 <= 1.5d-190) then
        tmp = 1.0d0
    else if ((y <= 2d-138) .or. (.not. (y <= 1.75d-122))) then
        tmp = (x - y) / y
    else
        tmp = 1.0d0 + ((-2.0d0) * ((y * y) / (x * x)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-190) {
		tmp = 1.0;
	} else if ((y <= 2e-138) || !(y <= 1.75e-122)) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.5e-190:
		tmp = 1.0
	elif (y <= 2e-138) or not (y <= 1.75e-122):
		tmp = (x - y) / y
	else:
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-190)
		tmp = 1.0;
	elseif ((y <= 2e-138) || !(y <= 1.75e-122))
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y * y) / Float64(x * x))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-190)
		tmp = 1.0;
	elseif ((y <= 2e-138) || ~((y <= 1.75e-122)))
		tmp = (x - y) / y;
	else
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.5e-190], 1.0, If[Or[LessEqual[y, 2e-138], N[Not[LessEqual[y, 1.75e-122]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4999999999999999e-190

   1. Initial program 61.7%

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

      1. +-commutative 61.7%

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

      2. +-commutative 61.7%

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

      3. associate-*r/ 61.9%

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

      4. +-commutative 61.9%

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

      5. +-commutative 61.9%

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

      6. fma-def 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in x around inf 37.7%

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

   if 1.4999999999999999e-190 < y < 2.00000000000000013e-138 or 1.7500000000000001e-122 < y

   1. Initial program 87.5%

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

      1. +-commutative 87.5%

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

      2. +-commutative 87.5%

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

      3. associate-*r/ 86.2%

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

      4. +-commutative 86.2%

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

      5. +-commutative 86.2%

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

      6. fma-def 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in x around 0 68.8%

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

   5. Taylor expanded in x around 0 68.8%

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

      1. *-inverses 68.8%

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

      2. div-sub 68.8%

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

   7. Simplified 68.8%

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

   if 2.00000000000000013e-138 < y < 1.7500000000000001e-122

   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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r/ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 75.8%

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

      1. unpow2 75.8%

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

      2. unpow2 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-138} \lor \neg \left(y \leq 1.75 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \end{array} \]

Alternative 5: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-189} \lor \neg \left(y \leq 1.2 \cdot 10^{-135}\right) \land y \leq 3.5 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -2}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (or (<= y 6e-189) (and (not (<= y 1.2e-135)) (<= y 3.5e-122)))
   (+ 1.0 (/ (/ (* y -2.0) (/ x y)) x))
   (/ (- x y) y)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((y <= 6e-189) || (!(y <= 1.2e-135) && (y <= 3.5e-122))) {
		tmp = 1.0 + (((y * -2.0) / (x / y)) / x);
	} else {
		tmp = (x - y) / y;
	}
	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 <= 6d-189) .or. (.not. (y <= 1.2d-135)) .and. (y <= 3.5d-122)) then
        tmp = 1.0d0 + (((y * (-2.0d0)) / (x / y)) / x)
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((y <= 6e-189) || (!(y <= 1.2e-135) && (y <= 3.5e-122))) {
		tmp = 1.0 + (((y * -2.0) / (x / y)) / x);
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if (y <= 6e-189) or (not (y <= 1.2e-135) and (y <= 3.5e-122)):
		tmp = 1.0 + (((y * -2.0) / (x / y)) / x)
	else:
		tmp = (x - y) / y
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if ((y <= 6e-189) || (!(y <= 1.2e-135) && (y <= 3.5e-122)))
		tmp = Float64(1.0 + Float64(Float64(Float64(y * -2.0) / Float64(x / y)) / x));
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 6e-189) || (~((y <= 1.2e-135)) && (y <= 3.5e-122)))
		tmp = 1.0 + (((y * -2.0) / (x / y)) / x);
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[Or[LessEqual[y, 6e-189], And[N[Not[LessEqual[y, 1.2e-135]], $MachinePrecision], LessEqual[y, 3.5e-122]]], N[(1.0 + N[(N[(N[(y * -2.0), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6e-189 or 1.1999999999999999e-135 < y < 3.5000000000000001e-122

   1. Initial program 62.5%

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

      1. +-commutative 62.5%

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

      2. +-commutative 62.5%

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

      3. associate-*r/ 62.6%

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

      4. +-commutative 62.6%

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

      5. +-commutative 62.6%

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

      6. fma-def 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in y around 0 28.7%

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

      1. unpow2 28.7%

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

      2. unpow2 28.7%

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

   6. Simplified 28.7%

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

      1. associate-/r* 39.7%

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

      2. associate-*r/ 39.7%

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

      3. associate-/l* 40.3%

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

      4. associate-*r/ 40.3%

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

      5. *-commutative 40.3%

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

   8. Applied egg-rr 40.3%

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

   if 6e-189 < y < 1.1999999999999999e-135 or 3.5000000000000001e-122 < y

   1. Initial program 87.5%

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

      1. +-commutative 87.5%

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

      2. +-commutative 87.5%

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

      3. associate-*r/ 86.2%

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

      4. +-commutative 86.2%

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

      5. +-commutative 86.2%

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

      6. fma-def 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in x around 0 68.8%

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

   5. Taylor expanded in x around 0 68.8%

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

      1. *-inverses 68.8%

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

      2. div-sub 68.8%

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

   7. Simplified 68.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-189} \lor \neg \left(y \leq 1.2 \cdot 10^{-135}\right) \land y \leq 3.5 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -2}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 6: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-134} \lor \neg \left(y \leq 1.8 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5e-189)
   1.0
   (if (or (<= y 1.55e-134) (not (<= y 1.8e-120))) (/ (- x y) y) 1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5e-189) {
		tmp = 1.0;
	} else if ((y <= 1.55e-134) || !(y <= 1.8e-120)) {
		tmp = (x - y) / y;
	} 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 <= 5d-189) then
        tmp = 1.0d0
    else if ((y <= 1.55d-134) .or. (.not. (y <= 1.8d-120))) then
        tmp = (x - y) / y
    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 <= 5e-189) {
		tmp = 1.0;
	} else if ((y <= 1.55e-134) || !(y <= 1.8e-120)) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5e-189:
		tmp = 1.0
	elif (y <= 1.55e-134) or not (y <= 1.8e-120):
		tmp = (x - y) / y
	else:
		tmp = 1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5e-189)
		tmp = 1.0;
	elseif ((y <= 1.55e-134) || !(y <= 1.8e-120))
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-189)
		tmp = 1.0;
	elseif ((y <= 1.55e-134) || ~((y <= 1.8e-120)))
		tmp = (x - y) / y;
	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, 5e-189], 1.0, If[Or[LessEqual[y, 1.55e-134], N[Not[LessEqual[y, 1.8e-120]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999997e-189 or 1.55000000000000003e-134 < y < 1.8000000000000001e-120

   1. Initial program 62.3%

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

      1. +-commutative 62.3%

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

      2. +-commutative 62.3%

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

      3. associate-*r/ 62.5%

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

      4. +-commutative 62.5%

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

      5. +-commutative 62.5%

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

      6. fma-def 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in x around inf 38.1%

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

   if 4.9999999999999997e-189 < y < 1.55000000000000003e-134 or 1.8000000000000001e-120 < y

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-*r/ 86.5%

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

      4. +-commutative 86.5%

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

      5. +-commutative 86.5%

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

      6. fma-def 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in x around 0 67.5%

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

   5. Taylor expanded in x around 0 67.6%

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

      1. *-inverses 67.6%

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

      2. div-sub 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-134} \lor \neg \left(y \leq 1.8 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 81.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-157}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e-189) 1.0 (if (<= y 8e-157) -1.0 (if (<= y 2e-122) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-189) {
		tmp = 1.0;
	} else if (y <= 8e-157) {
		tmp = -1.0;
	} else if (y <= 2e-122) {
		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 <= 6.8d-189) then
        tmp = 1.0d0
    else if (y <= 8d-157) then
        tmp = -1.0d0
    else if (y <= 2d-122) 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 <= 6.8e-189) {
		tmp = 1.0;
	} else if (y <= 8e-157) {
		tmp = -1.0;
	} else if (y <= 2e-122) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 6.8e-189:
		tmp = 1.0
	elif y <= 8e-157:
		tmp = -1.0
	elif y <= 2e-122:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-189)
		tmp = 1.0;
	elseif (y <= 8e-157)
		tmp = -1.0;
	elseif (y <= 2e-122)
		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 <= 6.8e-189)
		tmp = 1.0;
	elseif (y <= 8e-157)
		tmp = -1.0;
	elseif (y <= 2e-122)
		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, 6.8e-189], 1.0, If[LessEqual[y, 8e-157], -1.0, If[LessEqual[y, 2e-122], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.8000000000000002e-189 or 7.99999999999999955e-157 < y < 2.00000000000000012e-122

   1. Initial program 63.4%

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

      1. +-commutative 63.4%

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

      2. +-commutative 63.4%

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

      3. associate-*r/ 63.5%

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

      4. +-commutative 63.5%

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

      5. +-commutative 63.5%

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

      6. fma-def 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in x around inf 39.4%

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

   if 6.8000000000000002e-189 < y < 7.99999999999999955e-157 or 2.00000000000000012e-122 < y

   1. Initial program 86.0%

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

      1. +-commutative 86.0%

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

      2. +-commutative 86.0%

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

      3. associate-*r/ 84.7%

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

      4. +-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-def 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in x around 0 72.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-157}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 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 67.2%

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

   1. +-commutative 67.2%

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

   2. +-commutative 67.2%

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

   3. associate-*r/ 67.1%

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

   4. +-commutative 67.1%

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

   5. +-commutative 67.1%

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

   6. fma-def 67.1%

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

3. Simplified 67.1%

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

4. Taylor expanded in x around 0 62.7%

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

5. Final simplification 62.7%

   \[\leadsto -1 \]

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 2023297 
(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))))