Kahan p9 Example

Percentage Accurate: 68.1% → 99.9%

Time: 8.8s

Alternatives: 11

Speedup: 0.1×

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: 68.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. add-sqr-sqrt 64.4%

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

   2. times-frac 65.5%

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

   3. hypot-define 65.5%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 47.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{x}}{\frac{x}{y}}, -2, 1\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \frac{{x}^{2} - y \cdot y}{x + y}}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.4e-177)
   (fma (/ (/ y x) (/ x y)) -2.0 1.0)
   (if (<= y 1.7e-47)
     (/ (* (+ x y) (/ (- (pow x 2.0) (* y y)) (+ x y))) (+ (* y y) (* x x)))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-177) {
		tmp = fma(((y / x) / (x / y)), -2.0, 1.0);
	} else if (y <= 1.7e-47) {
		tmp = ((x + y) * ((pow(x, 2.0) - (y * y)) / (x + y))) / ((y * y) + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.4e-177)
		tmp = fma(Float64(Float64(y / x) / Float64(x / y)), -2.0, 1.0);
	elseif (y <= 1.7e-47)
		tmp = Float64(Float64(Float64(x + y) * Float64(Float64((x ^ 2.0) - Float64(y * y)) / Float64(x + y))) / Float64(Float64(y * y) + Float64(x * x)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.4e-177], N[(N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], If[LessEqual[y, 1.7e-47], N[(N[(N[(x + y), $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4000000000000004e-177

   1. Initial program 58.2%

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

      1. add-sqr-sqrt 58.2%

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

      2. times-frac 59.5%

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

      3. hypot-define 59.5%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-times 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. fma-define 25.3%

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

      4. unpow2 25.3%

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

      5. unpow2 25.3%

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

      6. times-frac 38.2%

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

      7. unpow2 38.2%

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

   9. Simplified 38.2%

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

       1. unpow2 38.2%

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

       2. clear-num 38.2%

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

       3. un-div-inv 38.2%

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

   11. Applied egg-rr 38.2%

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

   if 5.4000000000000004e-177 < y < 1.7000000000000001e-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. sub-neg 100.0%

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

      2. flip-+ 100.0%

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

      3. pow2 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 1.7000000000000001e-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 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{x}}{\frac{x}{y}}, -2, 1\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \frac{{x}^{2} - y \cdot y}{x + y}}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{x}}{\frac{x}{y}}, -2, 1\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.4e-177)
   (fma (/ (/ y x) (/ x y)) -2.0 1.0)
   (if (<= y 1.7e-47) (/ (* (- x y) (+ x y)) (+ (* y y) (* x x))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-177) {
		tmp = fma(((y / x) / (x / y)), -2.0, 1.0);
	} else if (y <= 1.7e-47) {
		tmp = ((x - y) * (x + y)) / ((y * y) + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.4e-177)
		tmp = fma(Float64(Float64(y / x) / Float64(x / y)), -2.0, 1.0);
	elseif (y <= 1.7e-47)
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(y * y) + Float64(x * x)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.4e-177], N[(N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], If[LessEqual[y, 1.7e-47], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4000000000000004e-177

   1. Initial program 58.2%

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

      1. add-sqr-sqrt 58.2%

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

      2. times-frac 59.5%

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

      3. hypot-define 59.5%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-times 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. fma-define 25.3%

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

      4. unpow2 25.3%

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

      5. unpow2 25.3%

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

      6. times-frac 38.2%

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

      7. unpow2 38.2%

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

   9. Simplified 38.2%

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

       1. unpow2 38.2%

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

       2. clear-num 38.2%

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

       3. un-div-inv 38.2%

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

   11. Applied egg-rr 38.2%

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

   if 5.4000000000000004e-177 < y < 1.7000000000000001e-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 1.7000000000000001e-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 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{x}}{\frac{x}{y}}, -2, 1\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(\frac{y}{x} + -1\right)}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.4e-177)
   (/ (- x y) (+ x (* y (+ (/ y x) -1.0))))
   (if (<= y 1.7e-47) (/ (* (- x y) (+ x y)) (+ (* y y) (* x x))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-177) {
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)));
	} else if (y <= 1.7e-47) {
		tmp = ((x - y) * (x + y)) / ((y * y) + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.4d-177) then
        tmp = (x - y) / (x + (y * ((y / x) + (-1.0d0))))
    else if (y <= 1.7d-47) then
        tmp = ((x - y) * (x + y)) / ((y * y) + (x * x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-177) {
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)));
	} else if (y <= 1.7e-47) {
		tmp = ((x - y) * (x + y)) / ((y * y) + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.4e-177:
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)))
	elif y <= 1.7e-47:
		tmp = ((x - y) * (x + y)) / ((y * y) + (x * x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.4e-177)
		tmp = Float64(Float64(x - y) / Float64(x + Float64(y * Float64(Float64(y / x) + -1.0))));
	elseif (y <= 1.7e-47)
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(y * y) + Float64(x * x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.4e-177)
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)));
	elseif (y <= 1.7e-47)
		tmp = ((x - y) * (x + y)) / ((y * y) + (x * x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.4e-177], N[(N[(x - y), $MachinePrecision] / N[(x + N[(y * N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e-47], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4000000000000004e-177

   1. Initial program 58.2%

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

      1. associate-/l* 59.3%

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

      2. +-commutative 59.3%

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

      3. fma-define 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in x around inf 37.9%

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

      1. clear-num 37.9%

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

      2. un-div-inv 38.0%

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

   7. Applied egg-rr 38.0%

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

   8. Taylor expanded in y around 0 37.4%

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

   if 5.4000000000000004e-177 < y < 1.7000000000000001e-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 1.7000000000000001e-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 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-177}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(\frac{y}{x} + -1\right)}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-158}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(\frac{y}{x} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{y}{-1 - \frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.05e-158)
   (/ (- x y) (+ x (* y (+ (/ y x) -1.0))))
   (/ (- y x) (/ y (- -1.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-158) {
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)));
	} else {
		tmp = (y - x) / (y / (-1.0 - (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.05d-158) then
        tmp = (x - y) / (x + (y * ((y / x) + (-1.0d0))))
    else
        tmp = (y - x) / (y / ((-1.0d0) - (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-158) {
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)));
	} else {
		tmp = (y - x) / (y / (-1.0 - (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.05e-158:
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)))
	else:
		tmp = (y - x) / (y / (-1.0 - (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.05e-158)
		tmp = Float64(Float64(x - y) / Float64(x + Float64(y * Float64(Float64(y / x) + -1.0))));
	else
		tmp = Float64(Float64(y - x) / Float64(y / Float64(-1.0 - Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.05e-158)
		tmp = (x - y) / (x + (y * ((y / x) + -1.0)));
	else
		tmp = (y - x) / (y / (-1.0 - (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.05e-158], N[(N[(x - y), $MachinePrecision] / N[(x + N[(y * N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] / N[(y / N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.05000000000000002e-158

   1. Initial program 58.6%

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

      1. associate-/l* 59.7%

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

      2. +-commutative 59.7%

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

      3. fma-define 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. clear-num 38.5%

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

      2. un-div-inv 38.6%

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

   7. Applied egg-rr 38.6%

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

   8. Taylor expanded in y around 0 38.0%

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

   if 2.05000000000000002e-158 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 86.7%

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

      1. clear-num 86.7%

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

      2. un-div-inv 86.7%

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

   7. Applied egg-rr 86.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-158}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(\frac{y}{x} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{y}{-1 - \frac{x}{y}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(\frac{y}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{y}{-1 - \frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-155)
   (* (/ (- x y) x) (+ (/ y x) 1.0))
   (/ (- y x) (/ y (- -1.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-155) {
		tmp = ((x - y) / x) * ((y / x) + 1.0);
	} else {
		tmp = (y - x) / (y / (-1.0 - (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-155) then
        tmp = ((x - y) / x) * ((y / x) + 1.0d0)
    else
        tmp = (y - x) / (y / ((-1.0d0) - (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-155) {
		tmp = ((x - y) / x) * ((y / x) + 1.0);
	} else {
		tmp = (y - x) / (y / (-1.0 - (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e-155:
		tmp = ((x - y) / x) * ((y / x) + 1.0)
	else:
		tmp = (y - x) / (y / (-1.0 - (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-155)
		tmp = Float64(Float64(Float64(x - y) / x) * Float64(Float64(y / x) + 1.0));
	else
		tmp = Float64(Float64(y - x) / Float64(y / Float64(-1.0 - Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-155)
		tmp = ((x - y) / x) * ((y / x) + 1.0);
	else
		tmp = (y - x) / (y / (-1.0 - (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e-155], N[(N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] / N[(y / N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.20000000000000021e-155

   1. Initial program 58.6%

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

      1. associate-/l* 59.7%

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

      2. +-commutative 59.7%

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

      3. fma-define 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. clear-num 38.5%

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

      2. un-div-inv 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/r/ 38.6%

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

   9. Applied egg-rr 38.6%

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

   if 9.20000000000000021e-155 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 86.7%

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

      1. clear-num 86.7%

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

      2. un-div-inv 86.7%

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

   7. Applied egg-rr 86.7%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(\frac{y}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{y}{-1 - \frac{x}{y}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(\frac{y}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{x}{y}}{y} \cdot \left(y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-155)
   (* (/ (- x y) x) (+ (/ y x) 1.0))
   (* (/ (- -1.0 (/ x y)) y) (- y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-155) {
		tmp = ((x - y) / x) * ((y / x) + 1.0);
	} else {
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.3d-155) then
        tmp = ((x - y) / x) * ((y / x) + 1.0d0)
    else
        tmp = (((-1.0d0) - (x / y)) / y) * (y - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-155) {
		tmp = ((x - y) / x) * ((y / x) + 1.0);
	} else {
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-155:
		tmp = ((x - y) / x) * ((y / x) + 1.0)
	else:
		tmp = ((-1.0 - (x / y)) / y) * (y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-155)
		tmp = Float64(Float64(Float64(x - y) / x) * Float64(Float64(y / x) + 1.0));
	else
		tmp = Float64(Float64(Float64(-1.0 - Float64(x / y)) / y) * Float64(y - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-155)
		tmp = ((x - y) / x) * ((y / x) + 1.0);
	else
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-155], N[(N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.29999999999999986e-155

   1. Initial program 58.6%

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

      1. associate-/l* 59.7%

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

      2. +-commutative 59.7%

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

      3. fma-define 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. clear-num 38.5%

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

      2. un-div-inv 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/r/ 38.6%

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

   9. Applied egg-rr 38.6%

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

   if 3.29999999999999986e-155 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 86.7%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - y}{x} \cdot \left(\frac{y}{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{x}{y}}{y} \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.75 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{y}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{x}{y}}{y} \cdot \left(y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.75e-156)
   (* (- x y) (/ (+ (/ y x) 1.0) x))
   (* (/ (- -1.0 (/ x y)) y) (- y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.75e-156) {
		tmp = (x - y) * (((y / x) + 1.0) / x);
	} else {
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.75d-156) then
        tmp = (x - y) * (((y / x) + 1.0d0) / x)
    else
        tmp = (((-1.0d0) - (x / y)) / y) * (y - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.75e-156) {
		tmp = (x - y) * (((y / x) + 1.0) / x);
	} else {
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.75e-156:
		tmp = (x - y) * (((y / x) + 1.0) / x)
	else:
		tmp = ((-1.0 - (x / y)) / y) * (y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.75e-156)
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(y / x) + 1.0) / x));
	else
		tmp = Float64(Float64(Float64(-1.0 - Float64(x / y)) / y) * Float64(y - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.75e-156)
		tmp = (x - y) * (((y / x) + 1.0) / x);
	else
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.75e-156], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7499999999999999e-156

   1. Initial program 58.6%

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

      1. associate-/l* 59.7%

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

      2. +-commutative 59.7%

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

      3. fma-define 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around inf 38.5%

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

   if 2.7499999999999999e-156 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 86.7%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.75 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{y}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{x}{y}}{y} \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.3 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{x}{y}}{y} \cdot \left(y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.3e-159) 1.0 (* (/ (- -1.0 (/ x y)) y) (- y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.3e-159) {
		tmp = 1.0;
	} else {
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.3d-159) then
        tmp = 1.0d0
    else
        tmp = (((-1.0d0) - (x / y)) / y) * (y - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.3e-159) {
		tmp = 1.0;
	} else {
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.3e-159:
		tmp = 1.0
	else:
		tmp = ((-1.0 - (x / y)) / y) * (y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.3e-159)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(-1.0 - Float64(x / y)) / y) * Float64(y - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.3e-159)
		tmp = 1.0;
	else
		tmp = ((-1.0 - (x / y)) / y) * (y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.3e-159], 1.0, N[(N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.30000000000000047e-159

   1. Initial program 58.6%

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

   3. Taylor expanded in x around inf 37.1%

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

   if 8.30000000000000047e-159 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 86.7%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.3 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{x}{y}}{y} \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-158}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.5e-158) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-158) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d-158) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-158) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e-158:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-158)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-158)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e-158], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e-158

   1. Initial program 58.6%

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

   3. Taylor expanded in x around inf 37.1%

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

   if 1.5e-158 < 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 86.3%

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

Alternative 11: 66.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in x around 0 66.9%

   \[\leadsto \color{blue}{-1} \]
4. 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 2024111 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (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))))