Kahan p9 Example

Percentage Accurate: 67.5% → 100.0%

Time: 7.8s

Alternatives: 11

Speedup: 0.5×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (/ (- x y) (hypot x y)) (/ 1.0 (/ (hypot x y) (+ x y)))))

C

double code(double x, double y) {
	return ((x - y) / hypot(x, y)) * (1.0 / (hypot(x, y) / (x + y)));
}

Java

public static double code(double x, double y) {
	return ((x - y) / Math.hypot(x, y)) * (1.0 / (Math.hypot(x, y) / (x + y)));
}

Python

def code(x, y):
	return ((x - y) / math.hypot(x, y)) * (1.0 / (math.hypot(x, y) / (x + y)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) / hypot(x, y)) * Float64(1.0 / Float64(hypot(x, y) / Float64(x + y))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) / hypot(x, y)) * (1.0 / (hypot(x, y) / (x + y)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.2%

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

   2. times-frac 67.7%

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

   3. hypot-def 67.7%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 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 67.2%

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

   1. add-sqr-sqrt 67.2%

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

   2. times-frac 67.7%

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

   3. hypot-def 67.7%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 92.4% accurate, 0.1× speedup

Math

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

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 (* (/ x y) (/ x y)) -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, ((x / y) * (x / y)), -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(Float64(x / y) * Float64(x / y)), -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[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $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-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 54.8%

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

      1. fma-neg 54.8%

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

      2. unpow2 54.8%

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

      3. unpow2 54.8%

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

      4. times-frac 78.1%

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

      5. metadata-eval 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 92.8%

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

Alternative 4: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ t_1 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(t_1 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
        (t_1 (* (/ x y) (/ x y))))
   (if (<= t_0 2.0) t_0 (+ t_1 (+ t_1 -1.0)))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double t_1 = (x / y) * (x / y);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (t_1 + -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) :: t_1
    real(8) :: tmp
    t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    t_1 = (x / y) * (x / y)
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = t_1 + (t_1 + (-1.0d0))
    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 t_1 = (x / y) * (x / y);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (t_1 + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	t_1 = (x / y) * (x / y)
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = t_1 + (t_1 + -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)))
	t_1 = Float64(Float64(x / y) * Float64(x / y))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(t_1 + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	t_1 = (x / y) * (x / y);
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = t_1 + (t_1 + -1.0);
	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]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(t$95$1 + N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 99.9%

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

   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. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 54.8%

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

      1. distribute-rgt1-in 54.8%

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

      2. metadata-eval 54.8%

         \[\leadsto \left(\frac{{x}^{2}}{{y}^{2}} + \color{blue}{0} \cdot \frac{x}{y}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      3. mul0-lft 54.8%

         \[\leadsto \left(\frac{{x}^{2}}{{y}^{2}} + \color{blue}{0}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      4. associate-+r- 54.8%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} + \left(0 - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right)\right)} \]

      5. neg-sub0 54.8%

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

      6. unpow2 54.8%

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

      7. unpow2 54.8%

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

      8. times-frac 54.8%

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

      9. +-commutative 54.8%

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

      10. distribute-neg-in 54.8%

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

      11. metadata-eval 54.8%

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

      12. mul-1-neg 54.8%

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

      13. remove-double-neg 54.8%

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

      14. unpow2 54.8%

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

      15. unpow2 54.8%

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

      16. times-frac 78.0%

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

   10. Simplified 78.0%

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

4. Final simplification 92.8%

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

Alternative 5: 92.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (/ (* (+ x y) (/ 1.0 (/ y (- x y)))) 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 = ((x + y) * (1.0 / (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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = ((x + y) * (1.0d0 / (y / (x - y)))) / 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 = ((x + y) * (1.0 / (y / (x - y)))) / 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 = ((x + y) * (1.0 / (y / (x - y)))) / 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(Float64(Float64(x + y) * Float64(1.0 / Float64(y / Float64(x - y)))) / 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 = ((x + y) * (1.0 / (y / (x - y)))) / 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[(N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around inf 3.1%

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

      1. unpow2 3.1%

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

   6. Simplified 3.1%

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

      1. associate-*r/ 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. times-frac 77.3%

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

      2. associate-*r/ 77.3%

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

   10. Simplified 77.3%

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

       1. clear-num 77.3%

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

       2. inv-pow 77.3%

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

   12. Applied egg-rr 77.3%

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

       1. unpow-1 77.3%

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

   14. Simplified 77.3%

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

4. Final simplification 92.5%

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

Alternative 6: 48.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;1 + \frac{y \cdot -2}{\frac{x}{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 0.005:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.6e-159)
   (+ 1.0 (/ (* y -2.0) (/ x (/ y x))))
   (if (<= y 0.005) (* (- x y) (/ (+ x y) (+ (* x x) (* y y)))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.6e-159) {
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)));
	} else if (y <= 0.005) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * 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 <= 8.6d-159) then
        tmp = 1.0d0 + ((y * (-2.0d0)) / (x / (y / x)))
    else if (y <= 0.005d0) then
        tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.6e-159) {
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)));
	} else if (y <= 0.005) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.6e-159:
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)))
	elif y <= 0.005:
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.6e-159)
		tmp = Float64(1.0 + Float64(Float64(y * -2.0) / Float64(x / Float64(y / x))));
	elseif (y <= 0.005)
		tmp = Float64(Float64(x - y) * Float64(Float64(x + y) / Float64(Float64(x * x) + Float64(y * y))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.6e-159)
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)));
	elseif (y <= 0.005)
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.6e-159], N[(1.0 + N[(N[(y * -2.0), $MachinePrecision] / N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.005], N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 8.6e-159

   1. Initial program 59.2%

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

      1. associate-*r/ 59.6%

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

      2. +-commutative 59.6%

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

      3. fma-def 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in y around 0 26.3%

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

      1. unpow2 26.3%

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

      2. unpow2 26.3%

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

   6. Simplified 26.3%

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

      1. times-frac 35.8%

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

      2. unpow2 35.8%

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

      3. *-commutative 35.8%

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

      4. unpow2 35.8%

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

      5. times-frac 26.3%

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

      6. associate-/l* 26.6%

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

      7. associate-*l/ 26.6%

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

      8. associate-/l* 35.8%

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

   8. Applied egg-rr 35.8%

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

   if 8.6e-159 < y < 0.0050000000000000001

   1. Initial program 99.9%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-def 99.2%

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

   3. Simplified 99.2%

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

      1. fma-udef 99.2%

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

   5. Applied egg-rr 99.2%

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

   if 0.0050000000000000001 < 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-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;1 + \frac{y \cdot -2}{\frac{x}{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 0.005:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 43.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-112}:\\ \;\;\;\;1 + \frac{y \cdot -2}{\frac{x}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y} - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-112)
   (+ 1.0 (/ (* y -2.0) (/ x (/ y x))))
   (/ (- (* x (/ x y)) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-112) {
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)));
	} else {
		tmp = ((x * (x / y)) - 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 <= 1.85d-112) then
        tmp = 1.0d0 + ((y * (-2.0d0)) / (x / (y / x)))
    else
        tmp = ((x * (x / y)) - y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-112) {
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)));
	} else {
		tmp = ((x * (x / y)) - y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e-112:
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)))
	else:
		tmp = ((x * (x / y)) - y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-112)
		tmp = Float64(1.0 + Float64(Float64(y * -2.0) / Float64(x / Float64(y / x))));
	else
		tmp = Float64(Float64(Float64(x * Float64(x / y)) - y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-112)
		tmp = 1.0 + ((y * -2.0) / (x / (y / x)));
	else
		tmp = ((x * (x / y)) - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e-112], N[(1.0 + N[(N[(y * -2.0), $MachinePrecision] / N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8499999999999999e-112

   1. Initial program 62.8%

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

      1. associate-*r/ 63.1%

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

      2. +-commutative 63.1%

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

      3. fma-def 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in y around 0 29.4%

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

      1. unpow2 29.4%

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

      2. unpow2 29.4%

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

   6. Simplified 29.4%

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

      1. times-frac 38.1%

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

      2. unpow2 38.1%

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

      3. *-commutative 38.1%

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

      4. unpow2 38.1%

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

      5. times-frac 29.4%

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

      6. associate-/l* 29.7%

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

      7. associate-*l/ 29.7%

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

      8. associate-/l* 38.1%

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

   8. Applied egg-rr 38.1%

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

   if 1.8499999999999999e-112 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 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 y around inf 79.8%

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

      1. unpow2 79.8%

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

   6. Simplified 79.8%

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

      1. associate-*r/ 80.3%

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

   8. Applied egg-rr 80.3%

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

      1. times-frac 80.3%

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

      2. associate-*r/ 80.3%

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

   10. Simplified 80.3%

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

       1. clear-num 80.3%

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

       2. inv-pow 80.3%

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

   12. Applied egg-rr 80.3%

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

       1. unpow-1 80.3%

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

   14. Simplified 80.3%

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

   15. Taylor expanded in y around 0 80.3%

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

       1. neg-mul-1 80.3%

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

       2. neg-mul-1 80.3%

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

       3. associate-+r+ 80.3%

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

       4. distribute-neg-in 80.3%

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

       5. unpow2 80.3%

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

       6. associate-*l/ 80.3%

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

       7. associate-/r/ 80.3%

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

       8. associate-+r+ 80.3%

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

       9. distribute-neg-in 80.3%

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

       10. associate-+r+ 80.3%

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

       11. +-commutative 80.3%

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

       12. associate-+r+ 80.3%

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

       13. neg-mul-1 80.3%

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

       14. distribute-lft1-in 80.3%

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

       15. metadata-eval 80.3%

           \[\leadsto \frac{\left(\color{blue}{0} \cdot x + \left(-y\right)\right) + \frac{x}{\frac{y}{x}}}{y} \]

       16. mul0-lft 80.3%

           \[\leadsto \frac{\left(\color{blue}{0} + \left(-y\right)\right) + \frac{x}{\frac{y}{x}}}{y} \]

       17. sub-neg 80.3%

           \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + \frac{x}{\frac{y}{x}}}{y} \]

       18. neg-sub0 80.3%

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

       19. associate-/r/ 80.3%

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

       20. associate-*l/ 80.3%

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

       21. unpow2 80.3%

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

   17. Simplified 80.3%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-112}:\\ \;\;\;\;1 + \frac{y \cdot -2}{\frac{x}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y} - y}{y}\\ \end{array} \]

Alternative 8: 41.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-112) 1.0 (+ (* (/ x y) (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-112) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) * (x / y)) + -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 <= 1.85d-112) then
        tmp = 1.0d0
    else
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-112) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e-112:
		tmp = 1.0
	else:
		tmp = ((x / y) * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-112)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-112)
		tmp = 1.0;
	else
		tmp = ((x / y) * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e-112], 1.0, N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8499999999999999e-112

   1. Initial program 62.8%

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

      1. associate-*r/ 63.1%

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

      2. +-commutative 63.1%

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

      3. fma-def 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in x around inf 36.5%

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

   if 1.8499999999999999e-112 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 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 y around inf 79.8%

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

      1. unpow2 79.8%

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

   6. Simplified 79.8%

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

      1. associate-*r/ 80.3%

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

   8. Applied egg-rr 80.3%

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

      1. times-frac 80.3%

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

   10. Simplified 80.3%

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

   11. Taylor expanded in x around 0 80.3%

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

       1. sub-neg 80.3%

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

       2. remove-double-neg 80.3%

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

       3. mul-1-neg 80.3%

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

       4. distribute-neg-in 80.3%

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

       5. +-commutative 80.3%

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

       6. distribute-neg-in 80.3%

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

       7. metadata-eval 80.3%

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

       8. mul-1-neg 80.3%

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

       9. remove-double-neg 80.3%

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

       10. unpow2 80.3%

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

       11. unpow2 80.3%

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

       12. times-frac 80.3%

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

   13. Simplified 80.3%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 9: 41.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y} - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-112) 1.0 (/ (- (* x (/ x y)) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-112) {
		tmp = 1.0;
	} else {
		tmp = ((x * (x / y)) - 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 <= 1.85d-112) then
        tmp = 1.0d0
    else
        tmp = ((x * (x / y)) - y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-112) {
		tmp = 1.0;
	} else {
		tmp = ((x * (x / y)) - y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e-112:
		tmp = 1.0
	else:
		tmp = ((x * (x / y)) - y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-112)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x * Float64(x / y)) - y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-112)
		tmp = 1.0;
	else
		tmp = ((x * (x / y)) - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e-112], 1.0, N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8499999999999999e-112

   1. Initial program 62.8%

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

      1. associate-*r/ 63.1%

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

      2. +-commutative 63.1%

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

      3. fma-def 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in x around inf 36.5%

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

   if 1.8499999999999999e-112 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 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 y around inf 79.8%

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

      1. unpow2 79.8%

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

   6. Simplified 79.8%

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

      1. associate-*r/ 80.3%

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

   8. Applied egg-rr 80.3%

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

      1. times-frac 80.3%

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

      2. associate-*r/ 80.3%

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

   10. Simplified 80.3%

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

       1. clear-num 80.3%

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

       2. inv-pow 80.3%

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

   12. Applied egg-rr 80.3%

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

       1. unpow-1 80.3%

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

   14. Simplified 80.3%

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

   15. Taylor expanded in y around 0 80.3%

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

       1. neg-mul-1 80.3%

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

       2. neg-mul-1 80.3%

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

       3. associate-+r+ 80.3%

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

       4. distribute-neg-in 80.3%

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

       5. unpow2 80.3%

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

       6. associate-*l/ 80.3%

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

       7. associate-/r/ 80.3%

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

       8. associate-+r+ 80.3%

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

       9. distribute-neg-in 80.3%

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

       10. associate-+r+ 80.3%

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

       11. +-commutative 80.3%

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

       12. associate-+r+ 80.3%

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

       13. neg-mul-1 80.3%

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

       14. distribute-lft1-in 80.3%

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

       15. metadata-eval 80.3%

           \[\leadsto \frac{\left(\color{blue}{0} \cdot x + \left(-y\right)\right) + \frac{x}{\frac{y}{x}}}{y} \]

       16. mul0-lft 80.3%

           \[\leadsto \frac{\left(\color{blue}{0} + \left(-y\right)\right) + \frac{x}{\frac{y}{x}}}{y} \]

       17. sub-neg 80.3%

           \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + \frac{x}{\frac{y}{x}}}{y} \]

       18. neg-sub0 80.3%

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

       19. associate-/r/ 80.3%

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

       20. associate-*l/ 80.3%

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

       21. unpow2 80.3%

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

   17. Simplified 80.3%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y} - y}{y}\\ \end{array} \]

Alternative 10: 41.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.6e-112) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-112) {
		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 <= 2.6d-112) 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 <= 2.6e-112) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-112:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-112)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e-112)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-112], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.59999999999999992e-112

   1. Initial program 62.8%

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

      1. associate-*r/ 63.1%

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

      2. +-commutative 63.1%

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

      3. fma-def 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in x around inf 36.5%

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

   if 2.59999999999999992e-112 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 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 0 79.3%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11: 66.7% 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 67.2%

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

   1. associate-*r/ 67.4%

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

   2. +-commutative 67.4%

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

   3. fma-def 67.4%

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

3. Simplified 67.4%

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

4. Taylor expanded in x around 0 65.8%

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

5. Final simplification 65.8%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023238 
(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))))