Kahan p9 Example

Percentage Accurate: 67.7% → 99.9%

Time: 10.5s

Alternatives: 12

Speedup: 1.6×

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: 67.7% 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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

   1. associate-/l* 73.5%

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

   2. fma-def 73.5%

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

3. Simplified 73.5%

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

   1. fma-def 73.5%

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

   2. div-inv 73.2%

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

   3. add-sqr-sqrt 73.1%

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

   4. associate-*l* 73.3%

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

   5. hypot-def 73.3%

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

   6. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 97.5%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{x + y}\right)\right)}} \]

   2. expm1-udef 97.5%

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

   3. un-div-inv 97.6%

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

7. Applied egg-rr 97.6%

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

   1. expm1-def 97.6%

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

   2. expm1-log1p 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 92.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (fma 2.0 (pow (/ x y) 2.0) -1.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

      1. associate-/l* 3.1%

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

      2. fma-def 3.1%

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

   3. Simplified 3.1%

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

      1. fma-def 3.1%

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

      2. add-sqr-sqrt 3.1%

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

      3. *-un-lft-identity 3.1%

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

      4. times-frac 3.1%

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

      5. hypot-def 3.1%

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

      6. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 52.2%

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

      1. fma-neg 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 73.6%

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

      5. unpow2 73.6%

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

      6. metadata-eval 73.6%

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

   8. Simplified 73.6%

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

4. Final simplification 92.8%

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

Alternative 3: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

   1. associate-*r/ 73.2%

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

   2. +-commutative 73.2%

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

   3. fma-def 73.2%

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

3. Simplified 73.2%

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

   1. fma-udef 73.2%

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

   2. +-commutative 73.2%

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

   3. *-un-lft-identity 73.2%

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

   4. add-sqr-sqrt 73.2%

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

   5. times-frac 73.3%

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

   6. hypot-def 73.3%

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

   7. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

   1. add-sqr-sqrt 73.0%

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

   2. times-frac 73.5%

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

   3. hypot-def 73.5%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.9%

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

   2. +-commutative 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 5: 92.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (-1.0d0) + ((x / y) * (x / y))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 0.0%

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

      1. unpow2 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. sub-neg 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. metadata-eval 52.2%

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

   7. Simplified 52.2%

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

      1. times-frac 73.4%

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

   9. Applied egg-rr 73.4%

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

4. Final simplification 92.8%

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

Alternative 6: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y} \cdot \frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* (/ x y) (/ x y)))))
   (if (<= y -1.3e-153)
     t_0
     (if (<= y 4.2e-179)
       (+ (/ y x) (- 1.0 (/ y x)))
       (if (<= y 5.1e-123)
         t_0
         (if (<= y 1.7e-108) 1.0 (* (/ (- x y) y) (/ (+ x y) y))))))))

C

double code(double x, double y) {
	double t_0 = -1.0 + ((x / y) * (x / y));
	double tmp;
	if (y <= -1.3e-153) {
		tmp = t_0;
	} else if (y <= 4.2e-179) {
		tmp = (y / x) + (1.0 - (y / x));
	} else if (y <= 5.1e-123) {
		tmp = t_0;
	} else if (y <= 1.7e-108) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) / y) * ((x + y) / 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 = (-1.0d0) + ((x / y) * (x / y))
    if (y <= (-1.3d-153)) then
        tmp = t_0
    else if (y <= 4.2d-179) then
        tmp = (y / x) + (1.0d0 - (y / x))
    else if (y <= 5.1d-123) then
        tmp = t_0
    else if (y <= 1.7d-108) then
        tmp = 1.0d0
    else
        tmp = ((x - y) / y) * ((x + y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + ((x / y) * (x / y));
	double tmp;
	if (y <= -1.3e-153) {
		tmp = t_0;
	} else if (y <= 4.2e-179) {
		tmp = (y / x) + (1.0 - (y / x));
	} else if (y <= 5.1e-123) {
		tmp = t_0;
	} else if (y <= 1.7e-108) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) / y) * ((x + y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + ((x / y) * (x / y))
	tmp = 0
	if y <= -1.3e-153:
		tmp = t_0
	elif y <= 4.2e-179:
		tmp = (y / x) + (1.0 - (y / x))
	elif y <= 5.1e-123:
		tmp = t_0
	elif y <= 1.7e-108:
		tmp = 1.0
	else:
		tmp = ((x - y) / y) * ((x + y) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)))
	tmp = 0.0
	if (y <= -1.3e-153)
		tmp = t_0;
	elseif (y <= 4.2e-179)
		tmp = Float64(Float64(y / x) + Float64(1.0 - Float64(y / x)));
	elseif (y <= 5.1e-123)
		tmp = t_0;
	elseif (y <= 1.7e-108)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x - y) / y) * Float64(Float64(x + y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + ((x / y) * (x / y));
	tmp = 0.0;
	if (y <= -1.3e-153)
		tmp = t_0;
	elseif (y <= 4.2e-179)
		tmp = (y / x) + (1.0 - (y / x));
	elseif (y <= 5.1e-123)
		tmp = t_0;
	elseif (y <= 1.7e-108)
		tmp = 1.0;
	else
		tmp = ((x - y) / y) * ((x + y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3e-153], t$95$0, If[LessEqual[y, 4.2e-179], N[(N[(y / x), $MachinePrecision] + N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.1e-123], t$95$0, If[LessEqual[y, 1.7e-108], 1.0, N[(N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.3000000000000001e-153 or 4.1999999999999997e-179 < y < 5.1000000000000001e-123

   1. Initial program 71.6%

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

   2. Taylor expanded in x around 0 61.5%

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

      1. unpow2 61.5%

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

   4. Simplified 61.5%

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

   5. Taylor expanded in x around 0 87.1%

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

      1. sub-neg 87.1%

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

      2. unpow2 87.1%

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

      3. unpow2 87.1%

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

      4. metadata-eval 87.1%

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

   7. Simplified 87.1%

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

      1. times-frac 89.9%

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

   9. Applied egg-rr 89.9%

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

   if -1.3000000000000001e-153 < y < 4.1999999999999997e-179

   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-*r/ 59.4%

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

      2. +-commutative 59.4%

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

      3. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. unsub-neg 84.3%

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

   6. Simplified 84.3%

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

   if 5.1000000000000001e-123 < y < 1.70000000000000001e-108

   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-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 100.0%

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

   if 1.70000000000000001e-108 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.5%

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

      1. unpow2 89.5%

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

   4. Simplified 89.5%

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

      1. times-frac 89.5%

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

   6. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\frac{x - y}{y} \cdot \frac{x + y}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-153}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-123}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y} \cdot \frac{x + y}{y}\\ \end{array} \]

Alternative 7: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-178}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y} \cdot \frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* (/ x y) (/ x y)))))
   (if (<= y -8.8e-154)
     t_0
     (if (<= y 2.8e-178)
       (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))
       (if (<= y 6e-123)
         t_0
         (if (<= y 1.9e-108) 1.0 (* (/ (- x y) y) (/ (+ x y) y))))))))

C

double code(double x, double y) {
	double t_0 = -1.0 + ((x / y) * (x / y));
	double tmp;
	if (y <= -8.8e-154) {
		tmp = t_0;
	} else if (y <= 2.8e-178) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else if (y <= 6e-123) {
		tmp = t_0;
	} else if (y <= 1.9e-108) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) / y) * ((x + y) / 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 = (-1.0d0) + ((x / y) * (x / y))
    if (y <= (-8.8d-154)) then
        tmp = t_0
    else if (y <= 2.8d-178) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else if (y <= 6d-123) then
        tmp = t_0
    else if (y <= 1.9d-108) then
        tmp = 1.0d0
    else
        tmp = ((x - y) / y) * ((x + y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + ((x / y) * (x / y));
	double tmp;
	if (y <= -8.8e-154) {
		tmp = t_0;
	} else if (y <= 2.8e-178) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else if (y <= 6e-123) {
		tmp = t_0;
	} else if (y <= 1.9e-108) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) / y) * ((x + y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + ((x / y) * (x / y))
	tmp = 0
	if y <= -8.8e-154:
		tmp = t_0
	elif y <= 2.8e-178:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	elif y <= 6e-123:
		tmp = t_0
	elif y <= 1.9e-108:
		tmp = 1.0
	else:
		tmp = ((x - y) / y) * ((x + y) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)))
	tmp = 0.0
	if (y <= -8.8e-154)
		tmp = t_0;
	elseif (y <= 2.8e-178)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (y <= 6e-123)
		tmp = t_0;
	elseif (y <= 1.9e-108)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x - y) / y) * Float64(Float64(x + y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + ((x / y) * (x / y));
	tmp = 0.0;
	if (y <= -8.8e-154)
		tmp = t_0;
	elseif (y <= 2.8e-178)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	elseif (y <= 6e-123)
		tmp = t_0;
	elseif (y <= 1.9e-108)
		tmp = 1.0;
	else
		tmp = ((x - y) / y) * ((x + y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.8e-154], t$95$0, If[LessEqual[y, 2.8e-178], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-123], t$95$0, If[LessEqual[y, 1.9e-108], 1.0, N[(N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.80000000000000029e-154 or 2.80000000000000019e-178 < y < 5.99999999999999968e-123

   1. Initial program 71.6%

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

   2. Taylor expanded in x around 0 61.5%

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

      1. unpow2 61.5%

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

   4. Simplified 61.5%

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

   5. Taylor expanded in x around 0 87.1%

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

      1. sub-neg 87.1%

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

      2. unpow2 87.1%

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

      3. unpow2 87.1%

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

      4. metadata-eval 87.1%

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

   7. Simplified 87.1%

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

      1. times-frac 89.9%

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

   9. Applied egg-rr 89.9%

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

   if -8.80000000000000029e-154 < y < 2.80000000000000019e-178

   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-*r/ 59.4%

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

      2. +-commutative 59.4%

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

      3. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in y around 0 58.6%

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

      1. unpow2 58.6%

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

      2. unpow2 58.6%

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

   6. Simplified 58.6%

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

      1. times-frac 85.6%

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

   8. Applied egg-rr 85.6%

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

   if 5.99999999999999968e-123 < y < 1.89999999999999987e-108

   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-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 100.0%

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

   if 1.89999999999999987e-108 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.5%

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

      1. unpow2 89.5%

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

   4. Simplified 89.5%

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

      1. times-frac 89.5%

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

   6. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\frac{x - y}{y} \cdot \frac{x + y}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-154}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-178}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-123}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y} \cdot \frac{x + y}{y}\\ \end{array} \]

Alternative 8: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{\frac{x}{y} + -1}}{x + y}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-180}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-123}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y} \cdot \frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e-153)
   (/ 1.0 (/ (/ y (+ (/ x y) -1.0)) (+ x y)))
   (if (<= y 1.2e-180)
     (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))
     (if (<= y 7.8e-123)
       (+ -1.0 (* (/ x y) (/ x y)))
       (if (<= y 2.2e-108) 1.0 (* (/ (- x y) y) (/ (+ x y) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e-153) {
		tmp = 1.0 / ((y / ((x / y) + -1.0)) / (x + y));
	} else if (y <= 1.2e-180) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else if (y <= 7.8e-123) {
		tmp = -1.0 + ((x / y) * (x / y));
	} else if (y <= 2.2e-108) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) / y) * ((x + y) / 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 <= (-1.2d-153)) then
        tmp = 1.0d0 / ((y / ((x / y) + (-1.0d0))) / (x + y))
    else if (y <= 1.2d-180) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else if (y <= 7.8d-123) then
        tmp = (-1.0d0) + ((x / y) * (x / y))
    else if (y <= 2.2d-108) then
        tmp = 1.0d0
    else
        tmp = ((x - y) / y) * ((x + y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e-153) {
		tmp = 1.0 / ((y / ((x / y) + -1.0)) / (x + y));
	} else if (y <= 1.2e-180) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else if (y <= 7.8e-123) {
		tmp = -1.0 + ((x / y) * (x / y));
	} else if (y <= 2.2e-108) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) / y) * ((x + y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e-153:
		tmp = 1.0 / ((y / ((x / y) + -1.0)) / (x + y))
	elif y <= 1.2e-180:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	elif y <= 7.8e-123:
		tmp = -1.0 + ((x / y) * (x / y))
	elif y <= 2.2e-108:
		tmp = 1.0
	else:
		tmp = ((x - y) / y) * ((x + y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e-153)
		tmp = Float64(1.0 / Float64(Float64(y / Float64(Float64(x / y) + -1.0)) / Float64(x + y)));
	elseif (y <= 1.2e-180)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (y <= 7.8e-123)
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	elseif (y <= 2.2e-108)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x - y) / y) * Float64(Float64(x + y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e-153)
		tmp = 1.0 / ((y / ((x / y) + -1.0)) / (x + y));
	elseif (y <= 1.2e-180)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	elseif (y <= 7.8e-123)
		tmp = -1.0 + ((x / y) * (x / y));
	elseif (y <= 2.2e-108)
		tmp = 1.0;
	else
		tmp = ((x - y) / y) * ((x + y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e-153], N[(1.0 / N[(N[(y / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-180], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e-123], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-108], 1.0, N[(N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.2000000000000001e-153

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 62.9%

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

      1. unpow2 62.9%

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

   4. Simplified 62.9%

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

      1. clear-num 62.9%

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

      2. inv-pow 62.9%

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

   6. Applied egg-rr 62.9%

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

      1. unpow-1 62.9%

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

      2. unpow2 62.9%

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

      3. associate-/r* 63.7%

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

      4. unpow2 63.7%

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

   8. Simplified 63.7%

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

      1. expm1-log1p-u 61.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot y}{x - y}\right)\right)}}{x + y}} \]

      2. expm1-udef 34.1%

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

      3. associate-/l* 58.9%

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

      4. div-sub 58.9%

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

      5. pow1 58.9%

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

      6. pow1 58.9%

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

      7. pow-div 58.9%

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

      8. metadata-eval 58.9%

         \[\leadsto \frac{1}{\frac{e^{\mathsf{log1p}\left(\frac{y}{\frac{x}{y} - {y}^{\color{blue}{0}}}\right)} - 1}{x + y}} \]

      9. metadata-eval 58.9%

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

   10. Applied egg-rr 58.9%

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

       1. expm1-def 86.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\frac{x}{y} - 1}\right)\right)}}{x + y}} \]

       2. expm1-log1p 91.7%

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

       3. sub-neg 91.7%

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

       4. metadata-eval 91.7%

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

   12. Simplified 91.7%

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

   if -1.2000000000000001e-153 < y < 1.1999999999999999e-180

   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-*r/ 59.4%

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

      2. +-commutative 59.4%

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

      3. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in y around 0 58.6%

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

      1. unpow2 58.6%

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

      2. unpow2 58.6%

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

   6. Simplified 58.6%

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

      1. times-frac 85.6%

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

   8. Applied egg-rr 85.6%

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

   if 1.1999999999999999e-180 < y < 7.79999999999999952e-123

   1. Initial program 75.0%

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

   2. Taylor expanded in x around 0 51.3%

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

      1. unpow2 51.3%

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

   4. Simplified 51.3%

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

   5. Taylor expanded in x around 0 51.3%

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

      1. sub-neg 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

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

      4. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

      1. times-frac 76.4%

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

   9. Applied egg-rr 76.4%

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

   if 7.79999999999999952e-123 < y < 2.2000000000000001e-108

   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-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 100.0%

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

   if 2.2000000000000001e-108 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.5%

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

      1. unpow2 89.5%

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

   4. Simplified 89.5%

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

      1. times-frac 89.5%

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

   6. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\frac{x - y}{y} \cdot \frac{x + y}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{\frac{x}{y} + -1}}{x + y}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-180}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-123}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y} \cdot \frac{x + y}{y}\\ \end{array} \]

Alternative 9: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-153} \lor \neg \left(y \leq 1.65 \cdot 10^{-179} \lor \neg \left(y \leq 7.8 \cdot 10^{-123}\right) \land y \leq 5.1 \cdot 10^{-108}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4e-153)
         (not
          (or (<= y 1.65e-179) (and (not (<= y 7.8e-123)) (<= y 5.1e-108)))))
   (+ -1.0 (* (/ x y) (/ x y)))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e-153) || !((y <= 1.65e-179) || (!(y <= 7.8e-123) && (y <= 5.1e-108)))) {
		tmp = -1.0 + ((x / y) * (x / y));
	} 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 <= (-1.4d-153)) .or. (.not. (y <= 1.65d-179) .or. (.not. (y <= 7.8d-123)) .and. (y <= 5.1d-108))) then
        tmp = (-1.0d0) + ((x / y) * (x / y))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e-153) || !((y <= 1.65e-179) || (!(y <= 7.8e-123) && (y <= 5.1e-108)))) {
		tmp = -1.0 + ((x / y) * (x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4e-153) or not ((y <= 1.65e-179) or (not (y <= 7.8e-123) and (y <= 5.1e-108))):
		tmp = -1.0 + ((x / y) * (x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4e-153) || !((y <= 1.65e-179) || (!(y <= 7.8e-123) && (y <= 5.1e-108))))
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.4e-153) || ~(((y <= 1.65e-179) || (~((y <= 7.8e-123)) && (y <= 5.1e-108)))))
		tmp = -1.0 + ((x / y) * (x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4e-153], N[Not[Or[LessEqual[y, 1.65e-179], And[N[Not[LessEqual[y, 7.8e-123]], $MachinePrecision], LessEqual[y, 5.1e-108]]]], $MachinePrecision]], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e-153 or 1.6499999999999999e-179 < y < 7.79999999999999952e-123 or 5.1000000000000002e-108 < y

   1. Initial program 77.9%

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

   2. Taylor expanded in x around 0 67.7%

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

      1. unpow2 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in x around 0 87.6%

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

      1. sub-neg 87.6%

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

      2. unpow2 87.6%

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

      3. unpow2 87.6%

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

      4. metadata-eval 87.6%

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

   7. Simplified 87.6%

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

      1. times-frac 89.8%

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

   9. Applied egg-rr 89.8%

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

   if -1.4e-153 < y < 1.6499999999999999e-179 or 7.79999999999999952e-123 < y < 5.1000000000000002e-108

   1. Initial program 61.3%

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

      1. associate-*r/ 62.0%

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

      2. +-commutative 62.0%

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

      3. fma-def 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around inf 85.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-153} \lor \neg \left(y \leq 1.65 \cdot 10^{-179} \lor \neg \left(y \leq 7.8 \cdot 10^{-123}\right) \land y \leq 5.1 \cdot 10^{-108}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-123} \lor \neg \left(y \leq 1.7 \cdot 10^{-108}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* (/ x y) (/ x y)))))
   (if (<= y -1.4e-153)
     t_0
     (if (<= y 7.3e-180)
       (+ (/ y x) (- 1.0 (/ y x)))
       (if (or (<= y 7.6e-123) (not (<= y 1.7e-108))) t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 + ((x / y) * (x / y));
	double tmp;
	if (y <= -1.4e-153) {
		tmp = t_0;
	} else if (y <= 7.3e-180) {
		tmp = (y / x) + (1.0 - (y / x));
	} else if ((y <= 7.6e-123) || !(y <= 1.7e-108)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + ((x / y) * (x / y))
    if (y <= (-1.4d-153)) then
        tmp = t_0
    else if (y <= 7.3d-180) then
        tmp = (y / x) + (1.0d0 - (y / x))
    else if ((y <= 7.6d-123) .or. (.not. (y <= 1.7d-108))) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + ((x / y) * (x / y));
	double tmp;
	if (y <= -1.4e-153) {
		tmp = t_0;
	} else if (y <= 7.3e-180) {
		tmp = (y / x) + (1.0 - (y / x));
	} else if ((y <= 7.6e-123) || !(y <= 1.7e-108)) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + ((x / y) * (x / y))
	tmp = 0
	if y <= -1.4e-153:
		tmp = t_0
	elif y <= 7.3e-180:
		tmp = (y / x) + (1.0 - (y / x))
	elif (y <= 7.6e-123) or not (y <= 1.7e-108):
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)))
	tmp = 0.0
	if (y <= -1.4e-153)
		tmp = t_0;
	elseif (y <= 7.3e-180)
		tmp = Float64(Float64(y / x) + Float64(1.0 - Float64(y / x)));
	elseif ((y <= 7.6e-123) || !(y <= 1.7e-108))
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + ((x / y) * (x / y));
	tmp = 0.0;
	if (y <= -1.4e-153)
		tmp = t_0;
	elseif (y <= 7.3e-180)
		tmp = (y / x) + (1.0 - (y / x));
	elseif ((y <= 7.6e-123) || ~((y <= 1.7e-108)))
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e-153], t$95$0, If[LessEqual[y, 7.3e-180], N[(N[(y / x), $MachinePrecision] + N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 7.6e-123], N[Not[LessEqual[y, 1.7e-108]], $MachinePrecision]], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e-153 or 7.2999999999999996e-180 < y < 7.59999999999999991e-123 or 1.70000000000000001e-108 < y

   1. Initial program 77.9%

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

   2. Taylor expanded in x around 0 67.7%

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

      1. unpow2 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in x around 0 87.6%

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

      1. sub-neg 87.6%

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

      2. unpow2 87.6%

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

      3. unpow2 87.6%

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

      4. metadata-eval 87.6%

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

   7. Simplified 87.6%

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

      1. times-frac 89.8%

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

   9. Applied egg-rr 89.8%

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

   if -1.4e-153 < y < 7.2999999999999996e-180

   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-*r/ 59.4%

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

      2. +-commutative 59.4%

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

      3. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. unsub-neg 84.3%

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

   6. Simplified 84.3%

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

   if 7.59999999999999991e-123 < y < 1.70000000000000001e-108

   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-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-153}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{y}{x} + \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-123} \lor \neg \left(y \leq 1.7 \cdot 10^{-108}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-123}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e-153)
   -1.0
   (if (<= y 8.5e-179)
     1.0
     (if (<= y 6e-123) -1.0 (if (<= y 2.8e-108) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e-153) {
		tmp = -1.0;
	} else if (y <= 8.5e-179) {
		tmp = 1.0;
	} else if (y <= 6e-123) {
		tmp = -1.0;
	} else if (y <= 2.8e-108) {
		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 <= (-1d-153)) then
        tmp = -1.0d0
    else if (y <= 8.5d-179) then
        tmp = 1.0d0
    else if (y <= 6d-123) then
        tmp = -1.0d0
    else if (y <= 2.8d-108) 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 <= -1e-153) {
		tmp = -1.0;
	} else if (y <= 8.5e-179) {
		tmp = 1.0;
	} else if (y <= 6e-123) {
		tmp = -1.0;
	} else if (y <= 2.8e-108) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e-153:
		tmp = -1.0
	elif y <= 8.5e-179:
		tmp = 1.0
	elif y <= 6e-123:
		tmp = -1.0
	elif y <= 2.8e-108:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e-153)
		tmp = -1.0;
	elseif (y <= 8.5e-179)
		tmp = 1.0;
	elseif (y <= 6e-123)
		tmp = -1.0;
	elseif (y <= 2.8e-108)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e-153)
		tmp = -1.0;
	elseif (y <= 8.5e-179)
		tmp = 1.0;
	elseif (y <= 6e-123)
		tmp = -1.0;
	elseif (y <= 2.8e-108)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e-153], -1.0, If[LessEqual[y, 8.5e-179], 1.0, If[LessEqual[y, 6e-123], -1.0, If[LessEqual[y, 2.8e-108], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000004e-153 or 8.49999999999999932e-179 < y < 5.99999999999999968e-123 or 2.8e-108 < y

   1. Initial program 77.9%

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

      1. associate-*r/ 77.8%

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

      2. +-commutative 77.8%

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

      3. fma-def 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in x around 0 89.3%

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

   if -1.00000000000000004e-153 < y < 8.49999999999999932e-179 or 5.99999999999999968e-123 < y < 2.8e-108

   1. Initial program 61.3%

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

      1. associate-*r/ 62.0%

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

      2. +-commutative 62.0%

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

      3. fma-def 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around inf 85.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-123}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12: 65.9% 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 73.0%

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

   1. associate-*r/ 73.2%

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

   2. +-commutative 73.2%

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

   3. fma-def 73.2%

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

3. Simplified 73.2%

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

4. Taylor expanded in x around 0 67.2%

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

5. Final simplification 67.2%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023194 
(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))))