Kahan p9 Example

Percentage Accurate: 68.5% → 99.9%

Time: 9.7s

Alternatives: 9

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: 68.5% 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, 0.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right)}\right)\right) \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (* (expm1 (log1p (/ (- x y) (hypot x y)))) (/ (+ x y) (hypot x y))))

C

y = abs(y);
double code(double x, double y) {
	return expm1(log1p(((x - y) / hypot(x, y)))) * ((x + y) / hypot(x, y));
}

Java

y = Math.abs(y);
public static double code(double x, double y) {
	return Math.expm1(Math.log1p(((x - y) / Math.hypot(x, y)))) * ((x + y) / Math.hypot(x, y));
}

Python

y = abs(y)
def code(x, y):
	return math.expm1(math.log1p(((x - y) / math.hypot(x, y)))) * ((x + y) / math.hypot(x, y))

Julia

y = abs(y)
function code(x, y)
	return Float64(expm1(log1p(Float64(Float64(x - y) / hypot(x, y)))) * Float64(Float64(x + y) / hypot(x, y)))
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(Exp[N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. add-sqr-sqrt 69.0%

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

   2. times-frac 69.2%

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

   3. hypot-def 69.2%

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

   4. hypot-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. expm1-log1p-u 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right)}\right)\right) \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]

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 69.0%

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

   1. add-sqr-sqrt 69.0%

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

   2. times-frac 69.2%

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

   3. hypot-def 69.2%

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

   4. hypot-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: 92.5% 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 99.8%

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

   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. associate-/l* 3.1%

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

      2. +-commutative 3.1%

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

      3. remove-double-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

      7. sub-neg 3.1%

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

      8. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 77.9%

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

4. Final simplification 93.1%

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

Alternative 4: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-123} \lor \neg \left(y \leq 1.35 \cdot 10^{-115}\right):\\ \;\;\;\;-1 + \frac{2}{\frac{y}{x \cdot \frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-167)
   1.0
   (if (or (<= y 1.65e-123) (not (<= y 1.35e-115)))
     (+ -1.0 (/ 2.0 (/ y (* x (/ x y)))))
     1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-167) {
		tmp = 1.0;
	} else if ((y <= 1.65e-123) || !(y <= 1.35e-115)) {
		tmp = -1.0 + (2.0 / (y / (x * (x / 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 <= 2.8d-167) then
        tmp = 1.0d0
    else if ((y <= 1.65d-123) .or. (.not. (y <= 1.35d-115))) then
        tmp = (-1.0d0) + (2.0d0 / (y / (x * (x / 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 <= 2.8e-167) {
		tmp = 1.0;
	} else if ((y <= 1.65e-123) || !(y <= 1.35e-115)) {
		tmp = -1.0 + (2.0 / (y / (x * (x / y))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2.8e-167:
		tmp = 1.0
	elif (y <= 1.65e-123) or not (y <= 1.35e-115):
		tmp = -1.0 + (2.0 / (y / (x * (x / y))))
	else:
		tmp = 1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-167)
		tmp = 1.0;
	elseif ((y <= 1.65e-123) || !(y <= 1.35e-115))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(y / Float64(x * Float64(x / y)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.8e-167)
		tmp = 1.0;
	elseif ((y <= 1.65e-123) || ~((y <= 1.35e-115)))
		tmp = -1.0 + (2.0 / (y / (x * (x / 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, 2.8e-167], 1.0, If[Or[LessEqual[y, 1.65e-123], N[Not[LessEqual[y, 1.35e-115]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 2.79999999999999986e-167 or 1.6500000000000001e-123 < y < 1.35e-115

   1. Initial program 64.4%

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

   2. Taylor expanded in x around inf 35.6%

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

   if 2.79999999999999986e-167 < y < 1.6500000000000001e-123 or 1.35e-115 < y

   1. Initial program 94.9%

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

      1. add-sqr-sqrt 94.9%

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

      2. times-frac 94.3%

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

      3. hypot-def 94.3%

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

      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 x around 0 87.6%

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

      1. sub-neg 87.6%

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

      2. metadata-eval 87.6%

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

      3. +-commutative 87.6%

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

      4. associate-*r/ 87.6%

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

      5. associate-/l* 87.6%

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

      6. unpow2 87.6%

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

      7. unpow2 87.6%

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

      8. times-frac 92.7%

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

      9. unpow2 92.7%

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

   6. Simplified 92.7%

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

      1. unpow2 92.7%

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

      2. clear-num 92.7%

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

      3. frac-times 92.7%

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

      4. *-un-lft-identity 92.7%

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

   8. Applied egg-rr 92.7%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-123} \lor \neg \left(y \leq 1.35 \cdot 10^{-115}\right):\\ \;\;\;\;-1 + \frac{2}{\frac{y}{x \cdot \frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{if}\;y \leq 1.65 \cdot 10^{-167}:\\ \;\;\;\;1 + -2 \cdot t_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-123} \lor \neg \left(y \leq 6.8 \cdot 10^{-115}\right):\\ \;\;\;\;-1 + \frac{2}{t_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (/ x y)))))
   (if (<= y 1.65e-167)
     (+ 1.0 (* -2.0 t_0))
     (if (or (<= y 1.8e-123) (not (<= y 6.8e-115)))
       (+ -1.0 (/ 2.0 t_0))
       1.0))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = y / (x * (x / y));
	double tmp;
	if (y <= 1.65e-167) {
		tmp = 1.0 + (-2.0 * t_0);
	} else if ((y <= 1.8e-123) || !(y <= 6.8e-115)) {
		tmp = -1.0 + (2.0 / t_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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x / y))
    if (y <= 1.65d-167) then
        tmp = 1.0d0 + ((-2.0d0) * t_0)
    else if ((y <= 1.8d-123) .or. (.not. (y <= 6.8d-115))) then
        tmp = (-1.0d0) + (2.0d0 / t_0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = y / (x * (x / y));
	double tmp;
	if (y <= 1.65e-167) {
		tmp = 1.0 + (-2.0 * t_0);
	} else if ((y <= 1.8e-123) || !(y <= 6.8e-115)) {
		tmp = -1.0 + (2.0 / t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = y / (x * (x / y))
	tmp = 0
	if y <= 1.65e-167:
		tmp = 1.0 + (-2.0 * t_0)
	elif (y <= 1.8e-123) or not (y <= 6.8e-115):
		tmp = -1.0 + (2.0 / t_0)
	else:
		tmp = 1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x / y)))
	tmp = 0.0
	if (y <= 1.65e-167)
		tmp = Float64(1.0 + Float64(-2.0 * t_0));
	elseif ((y <= 1.8e-123) || !(y <= 6.8e-115))
		tmp = Float64(-1.0 + Float64(2.0 / t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = y / (x * (x / y));
	tmp = 0.0;
	if (y <= 1.65e-167)
		tmp = 1.0 + (-2.0 * t_0);
	elseif ((y <= 1.8e-123) || ~((y <= 6.8e-115)))
		tmp = -1.0 + (2.0 / t_0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.65e-167], N[(1.0 + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.8e-123], N[Not[LessEqual[y, 6.8e-115]], $MachinePrecision]], N[(-1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.64999999999999998e-167

   1. Initial program 64.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. times-frac 64.4%

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

      3. hypot-def 64.4%

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

      4. hypot-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 29.0%

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

      1. unpow2 29.0%

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

      2. unpow2 29.0%

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

      3. times-frac 36.8%

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

      4. unpow2 36.8%

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

   6. Simplified 36.8%

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

      1. unpow2 66.5%

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

      2. clear-num 66.5%

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

      3. frac-times 66.5%

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

      4. *-un-lft-identity 66.5%

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

   8. Applied egg-rr 36.8%

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

   if 1.64999999999999998e-167 < y < 1.7999999999999998e-123 or 6.7999999999999996e-115 < y

   1. Initial program 94.9%

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

      1. add-sqr-sqrt 94.9%

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

      2. times-frac 94.3%

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

      3. hypot-def 94.3%

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

      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 x around 0 87.6%

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

      1. sub-neg 87.6%

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

      2. metadata-eval 87.6%

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

      3. +-commutative 87.6%

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

      4. associate-*r/ 87.6%

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

      5. associate-/l* 87.6%

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

      6. unpow2 87.6%

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

      7. unpow2 87.6%

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

      8. times-frac 92.7%

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

      9. unpow2 92.7%

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

   6. Simplified 92.7%

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

      1. unpow2 92.7%

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

      2. clear-num 92.7%

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

      3. frac-times 92.7%

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

      4. *-un-lft-identity 92.7%

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

   8. Applied egg-rr 92.7%

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

   if 1.7999999999999998e-123 < y < 6.7999999999999996e-115

   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 inf 100.0%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-167}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-123} \lor \neg \left(y \leq 6.8 \cdot 10^{-115}\right):\\ \;\;\;\;-1 + \frac{2}{\frac{y}{x \cdot \frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 83.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-116}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - y}}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-167)
   1.0
   (if (<= y 4.9e-126)
     (/ (- x y) y)
     (if (<= y 3.6e-116) 1.0 (/ 1.0 (/ y (- x y)))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-167) {
		tmp = 1.0;
	} else if (y <= 4.9e-126) {
		tmp = (x - y) / y;
	} else if (y <= 3.6e-116) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (y / (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) :: tmp
    if (y <= 5.5d-167) then
        tmp = 1.0d0
    else if (y <= 4.9d-126) then
        tmp = (x - y) / y
    else if (y <= 3.6d-116) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (y / (x - y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-167) {
		tmp = 1.0;
	} else if (y <= 4.9e-126) {
		tmp = (x - y) / y;
	} else if (y <= 3.6e-116) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (y / (x - y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5.5e-167:
		tmp = 1.0
	elif y <= 4.9e-126:
		tmp = (x - y) / y
	elif y <= 3.6e-116:
		tmp = 1.0
	else:
		tmp = 1.0 / (y / (x - y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-167)
		tmp = 1.0;
	elseif (y <= 4.9e-126)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 3.6e-116)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(y / Float64(x - y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-167)
		tmp = 1.0;
	elseif (y <= 4.9e-126)
		tmp = (x - y) / y;
	elseif (y <= 3.6e-116)
		tmp = 1.0;
	else
		tmp = 1.0 / (y / (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5.5e-167], 1.0, If[LessEqual[y, 4.9e-126], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 3.6e-116], 1.0, N[(1.0 / N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.5000000000000003e-167 or 4.9000000000000001e-126 < y < 3.59999999999999975e-116

   1. Initial program 64.4%

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

   2. Taylor expanded in x around inf 35.6%

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

   if 5.5000000000000003e-167 < y < 4.9000000000000001e-126

   1. Initial program 81.8%

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

      1. associate-/l* 79.8%

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

      2. +-commutative 79.8%

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

      3. remove-double-neg 79.8%

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

      4. sub-neg 79.8%

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

      5. +-commutative 79.8%

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

      6. fma-def 79.8%

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

      7. sub-neg 79.8%

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

      8. remove-double-neg 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in x around 0 82.9%

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

   if 3.59999999999999975e-116 < 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* 99.8%

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

      2. +-commutative 99.8%

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

      3. remove-double-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. sub-neg 99.8%

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

      8. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 95.2%

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

      1. add-cbrt-cube 95.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x - y}{y} \cdot \frac{x - y}{y}\right) \cdot \frac{x - y}{y}}} \]

   6. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x - y}{y} \cdot \frac{x - y}{y}\right) \cdot \frac{x - y}{y}}} \]
   7. Step-by-step derivation

      1. associate-*l* 95.2%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{x - y}{y} \cdot \left(\frac{x - y}{y} \cdot \frac{x - y}{y}\right)}} \]

   8. Simplified 95.2%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{x - y}{y} \cdot \left(\frac{x - y}{y} \cdot \frac{x - y}{y}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 95.2%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{x - y}{y} \cdot \frac{x - y}{y}\right) \cdot \frac{x - y}{y}}} \]

      2. add-cbrt-cube 95.2%

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

      3. clear-num 95.2%

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

   10. Applied egg-rr 95.2%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-116}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - y}}\\ \end{array} \]

Alternative 7: 83.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-124} \lor \neg \left(y \leq 3.6 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e-168)
   1.0
   (if (or (<= y 9e-124) (not (<= y 3.6e-116))) (/ (- x y) y) 1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-168) {
		tmp = 1.0;
	} else if ((y <= 9e-124) || !(y <= 3.6e-116)) {
		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 <= 1.6d-168) then
        tmp = 1.0d0
    else if ((y <= 9d-124) .or. (.not. (y <= 3.6d-116))) 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 <= 1.6e-168) {
		tmp = 1.0;
	} else if ((y <= 9e-124) || !(y <= 3.6e-116)) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.6e-168:
		tmp = 1.0
	elif (y <= 9e-124) or not (y <= 3.6e-116):
		tmp = (x - y) / y
	else:
		tmp = 1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-168)
		tmp = 1.0;
	elseif ((y <= 9e-124) || !(y <= 3.6e-116))
		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 <= 1.6e-168)
		tmp = 1.0;
	elseif ((y <= 9e-124) || ~((y <= 3.6e-116)))
		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, 1.6e-168], 1.0, If[Or[LessEqual[y, 9e-124], N[Not[LessEqual[y, 3.6e-116]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000003e-168 or 8.9999999999999992e-124 < y < 3.59999999999999975e-116

   1. Initial program 64.4%

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

   2. Taylor expanded in x around inf 35.6%

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

   if 1.60000000000000003e-168 < y < 8.9999999999999992e-124 or 3.59999999999999975e-116 < y

   1. Initial program 94.9%

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

      1. associate-/l* 94.2%

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

      2. +-commutative 94.2%

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

      3. remove-double-neg 94.2%

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

      4. sub-neg 94.2%

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

      5. +-commutative 94.2%

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

      6. fma-def 94.2%

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

      7. sub-neg 94.2%

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

      8. remove-double-neg 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in x around 0 91.7%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-124} \lor \neg \left(y \leq 3.6 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 82.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-168)
   1.0
   (if (<= y 8e-134) -1.0 (if (<= y 4.8e-115) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-168) {
		tmp = 1.0;
	} else if (y <= 8e-134) {
		tmp = -1.0;
	} else if (y <= 4.8e-115) {
		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 <= 2.8d-168) then
        tmp = 1.0d0
    else if (y <= 8d-134) then
        tmp = -1.0d0
    else if (y <= 4.8d-115) 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 <= 2.8e-168) {
		tmp = 1.0;
	} else if (y <= 8e-134) {
		tmp = -1.0;
	} else if (y <= 4.8e-115) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2.8e-168:
		tmp = 1.0
	elif y <= 8e-134:
		tmp = -1.0
	elif y <= 4.8e-115:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-168)
		tmp = 1.0;
	elseif (y <= 8e-134)
		tmp = -1.0;
	elseif (y <= 4.8e-115)
		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 <= 2.8e-168)
		tmp = 1.0;
	elseif (y <= 8e-134)
		tmp = -1.0;
	elseif (y <= 4.8e-115)
		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, 2.8e-168], 1.0, If[LessEqual[y, 8e-134], -1.0, If[LessEqual[y, 4.8e-115], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8000000000000002e-168 or 8.00000000000000032e-134 < y < 4.80000000000000042e-115

   1. Initial program 64.5%

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

   2. Taylor expanded in x around inf 35.4%

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

   if 2.8000000000000002e-168 < y < 8.00000000000000032e-134 or 4.80000000000000042e-115 < y

   1. Initial program 94.7%

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

   2. Taylor expanded in x around 0 92.2%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 66.5% 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 69.0%

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

2. Taylor expanded in x around 0 68.8%

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

3. Final simplification 68.8%

   \[\leadsto -1 \]

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 2023275 
(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))))