Kahan p9 Example

Percentage Accurate: 68.7% → 100.0%

Time: 12.2s

Alternatives: 11

Speedup: 2.5×

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: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (/ (+ x y_m) (hypot x y_m)) (/ (hypot x y_m) (- x y_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x - y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.3%

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

   1. add-sqr-sqrt 63.3%

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

   2. times-frac 64.1%

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

   3. hypot-def 64.1%

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

   4. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(1 - \frac{y_m}{x}\right)}{\mathsf{hypot}\left(x, y_m\right)}\\ \mathbf{elif}\;y_m \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{{\left(\mathsf{hypot}\left(x, y_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (/ (* (+ x y_m) (- 1.0 (/ y_m x))) (hypot x y_m))
   (if (<= y_m 4.2e-47)
     (/ (* (+ x y_m) (- x y_m)) (pow (hypot x y_m) 2.0))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / hypot(x, y_m);
	} else if (y_m <= 4.2e-47) {
		tmp = ((x + y_m) * (x - y_m)) / pow(hypot(x, y_m), 2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / Math.hypot(x, y_m);
	} else if (y_m <= 4.2e-47) {
		tmp = ((x + y_m) * (x - y_m)) / Math.pow(Math.hypot(x, y_m), 2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.55e-162:
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / math.hypot(x, y_m)
	elif y_m <= 4.2e-47:
		tmp = ((x + y_m) * (x - y_m)) / math.pow(math.hypot(x, y_m), 2.0)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.55e-162)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(1.0 - Float64(y_m / x))) / hypot(x, y_m));
	elseif (y_m <= 4.2e-47)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / (hypot(x, y_m) ^ 2.0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.55e-162)
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / hypot(x, y_m);
	elseif (y_m <= 4.2e-47)
		tmp = ((x + y_m) * (x - y_m)) / (hypot(x, y_m) ^ 2.0);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.55e-162], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 4.2e-47], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 56.3%

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

      1. add-sqr-sqrt 56.3%

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

      2. times-frac 57.3%

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

      3. hypot-def 57.3%

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

      4. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 30.7%

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

      1. neg-mul-1 30.7%

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

      2. sub-neg 30.7%

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

   7. Simplified 30.7%

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

      1. associate-*r/ 30.7%

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

   9. Applied egg-rr 30.7%

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

   if 1.5499999999999999e-162 < y < 4.2000000000000001e-47

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 6.5%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{e^{\mathsf{log1p}\left(x \cdot x + y \cdot y\right)} - 1}} \]

      3. add-sqr-sqrt 6.5%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}\right)} - 1} \]

      4. pow2 6.5%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{x \cdot x + y \cdot y}\right)}^{2}}\right)} - 1} \]

      5. hypot-def 6.5%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2}\right)} - 1} \]

   4. Applied egg-rr 6.5%

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}\right)\right)}} \]

      2. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   if 4.2000000000000001e-47 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(1 - \frac{y}{x}\right)}{\mathsf{hypot}\left(x, y\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (/ (+ x y_m) (hypot x y_m)) (/ (- x y_m) (hypot x y_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.3%

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

   1. add-sqr-sqrt 63.3%

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

   2. times-frac 64.1%

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

   3. hypot-def 64.1%

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

   4. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x + y_m}{\mathsf{hypot}\left(x, y_m\right)} \cdot \left(1 - \frac{y_m}{x}\right)\\ \mathbf{elif}\;y_m \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (* (/ (+ x y_m) (hypot x y_m)) (- 1.0 (/ y_m x)))
   (if (<= y_m 4.25e-47)
     (/ (* (+ x y_m) (- x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) / hypot(x, y_m)) * (1.0 - (y_m / x));
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) / Math.hypot(x, y_m)) * (1.0 - (y_m / x));
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.55e-162:
		tmp = ((x + y_m) / math.hypot(x, y_m)) * (1.0 - (y_m / x))
	elif y_m <= 4.25e-47:
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.55e-162)
		tmp = Float64(Float64(Float64(x + y_m) / hypot(x, y_m)) * Float64(1.0 - Float64(y_m / x)));
	elseif (y_m <= 4.25e-47)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.55e-162)
		tmp = ((x + y_m) / hypot(x, y_m)) * (1.0 - (y_m / x));
	elseif (y_m <= 4.25e-47)
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.55e-162], N[(N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 4.25e-47], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 56.3%

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

      1. add-sqr-sqrt 56.3%

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

      2. times-frac 57.3%

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

      3. hypot-def 57.3%

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

      4. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 30.7%

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

      1. neg-mul-1 30.7%

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

      2. sub-neg 30.7%

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

   7. Simplified 30.7%

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

   if 1.5499999999999999e-162 < y < 4.24999999999999995e-47

   1. Initial program 100.0%

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

   if 4.24999999999999995e-47 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(1 - \frac{y_m}{x}\right)}{\mathsf{hypot}\left(x, y_m\right)}\\ \mathbf{elif}\;y_m \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (/ (* (+ x y_m) (- 1.0 (/ y_m x))) (hypot x y_m))
   (if (<= y_m 4.25e-47)
     (/ (* (+ x y_m) (- x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / hypot(x, y_m);
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / Math.hypot(x, y_m);
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.55e-162:
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / math.hypot(x, y_m)
	elif y_m <= 4.25e-47:
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.55e-162)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(1.0 - Float64(y_m / x))) / hypot(x, y_m));
	elseif (y_m <= 4.25e-47)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.55e-162)
		tmp = ((x + y_m) * (1.0 - (y_m / x))) / hypot(x, y_m);
	elseif (y_m <= 4.25e-47)
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.55e-162], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 4.25e-47], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 56.3%

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

      1. add-sqr-sqrt 56.3%

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

      2. times-frac 57.3%

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

      3. hypot-def 57.3%

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

      4. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 30.7%

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

      1. neg-mul-1 30.7%

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

      2. sub-neg 30.7%

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

   7. Simplified 30.7%

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

      1. associate-*r/ 30.7%

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

   9. Applied egg-rr 30.7%

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

   if 1.5499999999999999e-162 < y < 4.24999999999999995e-47

   1. Initial program 100.0%

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

   if 4.24999999999999995e-47 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(1 - \frac{y}{x}\right)}{\mathsf{hypot}\left(x, y\right)}\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;1 + -2 \cdot {\left(\frac{y_m}{x}\right)}^{2}\\ \mathbf{elif}\;y_m \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (+ 1.0 (* -2.0 (pow (/ y_m x) 2.0)))
   (if (<= y_m 4.25e-47)
     (/ (* (+ x y_m) (- x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = 1.0 + (-2.0 * pow((y_m / x), 2.0));
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.55d-162) then
        tmp = 1.0d0 + ((-2.0d0) * ((y_m / x) ** 2.0d0))
    else if (y_m <= 4.25d-47) then
        tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = 1.0 + (-2.0 * Math.pow((y_m / x), 2.0));
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.55e-162:
		tmp = 1.0 + (-2.0 * math.pow((y_m / x), 2.0))
	elif y_m <= 4.25e-47:
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.55e-162)
		tmp = Float64(1.0 + Float64(-2.0 * (Float64(y_m / x) ^ 2.0)));
	elseif (y_m <= 4.25e-47)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.55e-162)
		tmp = 1.0 + (-2.0 * ((y_m / x) ^ 2.0));
	elseif (y_m <= 4.25e-47)
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.55e-162], N[(1.0 + N[(-2.0 * N[Power[N[(y$95$m / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 4.25e-47], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 56.3%

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

      1. associate-/l* 57.3%

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

      2. remove-double-neg 57.3%

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

      3. sub-neg 57.3%

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

      4. +-commutative 57.3%

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

      5. fma-def 57.3%

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

      6. sub-neg 57.3%

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

      7. remove-double-neg 57.3%

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

   3. Simplified 57.3%

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

      1. div-inv 57.1%

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

      2. fma-udef 57.1%

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

      3. +-commutative 57.1%

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

      4. add-sqr-sqrt 57.1%

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

      5. associate-*l* 57.2%

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

      6. hypot-def 57.2%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. add-sqr-sqrt 31.4%

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

      2. pow2 31.4%

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

      3. associate-*r* 16.3%

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

      4. unpow2 16.3%

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

      5. sqrt-prod 16.3%

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

      6. unpow2 16.3%

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

      7. sqrt-prod 31.2%

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

      8. add-sqr-sqrt 31.3%

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

      9. inv-pow 31.3%

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

      10. sqrt-pow1 31.3%

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

      11. metadata-eval 31.3%

          \[\leadsto \frac{x - y}{{\left(\mathsf{hypot}\left(x, y\right) \cdot {\left(x + y\right)}^{\color{blue}{-0.5}}\right)}^{2}} \]

   8. Applied egg-rr 31.3%

      \[\leadsto \frac{x - y}{\color{blue}{{\left(\mathsf{hypot}\left(x, y\right) \cdot {\left(x + y\right)}^{-0.5}\right)}^{2}}} \]

   9. Taylor expanded in y around 0 18.6%

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

       1. unpow2 18.6%

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

       2. unpow2 18.6%

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

       3. times-frac 30.4%

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

       4. unpow2 30.4%

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

   11. Simplified 30.4%

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

   if 1.5499999999999999e-162 < y < 4.24999999999999995e-47

   1. Initial program 100.0%

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

   if 4.24999999999999995e-47 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;1 + -2 \cdot {\left(\frac{y}{x}\right)}^{2}\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(1 - \frac{y_m}{x}\right) \cdot \left(1 + \frac{y_m}{x}\right)\\ \mathbf{elif}\;y_m \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (* (- 1.0 (/ y_m x)) (+ 1.0 (/ y_m x)))
   (if (<= y_m 4.25e-47)
     (/ (* (+ x y_m) (- x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.55d-162) then
        tmp = (1.0d0 - (y_m / x)) * (1.0d0 + (y_m / x))
    else if (y_m <= 4.25d-47) then
        tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else if (y_m <= 4.25e-47) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.55e-162:
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x))
	elif y_m <= 4.25e-47:
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.55e-162)
		tmp = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(1.0 + Float64(y_m / x)));
	elseif (y_m <= 4.25e-47)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.55e-162)
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	elseif (y_m <= 4.25e-47)
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.55e-162], N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 4.25e-47], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 56.3%

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

      1. add-sqr-sqrt 56.3%

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

      2. times-frac 57.3%

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

      3. hypot-def 57.3%

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

      4. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 30.7%

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

      1. neg-mul-1 30.7%

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

      2. sub-neg 30.7%

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

   7. Simplified 30.7%

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

   8. Taylor expanded in x around inf 30.1%

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

   if 1.5499999999999999e-162 < y < 4.24999999999999995e-47

   1. Initial program 100.0%

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

   if 4.24999999999999995e-47 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 8.5 \cdot 10^{-174}:\\ \;\;\;\;\left(1 - \frac{y_m}{x}\right) \cdot \left(1 + \frac{y_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 8.5e-174) (* (- 1.0 (/ y_m x)) (+ 1.0 (/ y_m x))) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.5e-174) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 8.5d-174) then
        tmp = (1.0d0 - (y_m / x)) * (1.0d0 + (y_m / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.5e-174) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 8.5e-174:
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 8.5e-174)
		tmp = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(1.0 + Float64(y_m / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 8.5e-174)
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 8.5e-174], N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 8.4999999999999996e-174

   1. Initial program 56.5%

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

      1. add-sqr-sqrt 56.5%

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

      2. times-frac 57.6%

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

      3. hypot-def 57.6%

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

      4. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 30.8%

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

      1. neg-mul-1 30.8%

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

      2. sub-neg 30.8%

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

   7. Simplified 30.8%

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

   8. Taylor expanded in x around inf 30.2%

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

   if 8.4999999999999996e-174 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0 71.9%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-174}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 3.3 \cdot 10^{-181}:\\ \;\;\;\;\left(1 - \frac{y_m}{x}\right) \cdot \left(1 + \frac{y_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \frac{x}{y_m}\right) \cdot \left(1 + \frac{x}{y_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.3e-181)
   (* (- 1.0 (/ y_m x)) (+ 1.0 (/ y_m x)))
   (* (+ -1.0 (/ x y_m)) (+ 1.0 (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.3e-181) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = (-1.0 + (x / y_m)) * (1.0 + (x / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 3.3d-181) then
        tmp = (1.0d0 - (y_m / x)) * (1.0d0 + (y_m / x))
    else
        tmp = ((-1.0d0) + (x / y_m)) * (1.0d0 + (x / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.3e-181) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = (-1.0 + (x / y_m)) * (1.0 + (x / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 3.3e-181:
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x))
	else:
		tmp = (-1.0 + (x / y_m)) * (1.0 + (x / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3.3e-181)
		tmp = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(1.0 + Float64(y_m / x)));
	else
		tmp = Float64(Float64(-1.0 + Float64(x / y_m)) * Float64(1.0 + Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 3.3e-181)
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	else
		tmp = (-1.0 + (x / y_m)) * (1.0 + (x / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 3.3e-181], N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.30000000000000009e-181

   1. Initial program 56.6%

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

      1. add-sqr-sqrt 56.6%

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

      2. times-frac 57.6%

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

      3. hypot-def 57.7%

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

      4. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 30.2%

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

      1. neg-mul-1 30.2%

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

      2. sub-neg 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in x around inf 29.6%

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

   if 3.30000000000000009e-181 < y

   1. Initial program 95.5%

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

      1. add-sqr-sqrt 95.5%

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

      2. times-frac 95.4%

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

      3. hypot-def 95.4%

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

      4. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 70.8%

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

   6. Taylor expanded in x around 0 70.3%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-181}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \frac{x}{y}\right) \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 4 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 4e-174) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4e-174) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 4d-174) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4e-174) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 4e-174:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 4e-174)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 4e-174)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 4e-174], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4e-174

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 28.4%

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

   if 4e-174 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0 71.9%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.7% accurate, 15.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = -1.0d0
end function

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in x around 0 71.6%

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

4. Final simplification 71.6%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(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))))