Kahan p9 Example

Percentage Accurate: 68.0% → 99.9%

Time: 8.8s

Alternatives: 12

Speedup: 0.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.0% 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

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

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) / Float64(hypot(x, y) / 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[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.0%

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

   1. add-sqr-sqrt 70.0%

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

   2. times-frac 70.4%

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

   3. hypot-define 70.4%

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

   4. hypot-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. clear-num 99.9%

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

   3. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 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 70.0%

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

   1. associate-/l* 70.2%

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

   2. +-commutative 70.2%

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

   3. fma-define 70.2%

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

3. Simplified 70.2%

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

   1. fma-undefine 70.2%

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

   2. +-commutative 70.2%

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

   3. add-cube-cbrt 68.9%

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

   4. add-sqr-sqrt 68.9%

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

   5. times-frac 68.9%

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

   6. pow2 68.9%

      \[\leadsto \left(x - y\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x + y}\right)}^{2}}}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{x \cdot x + y \cdot y}}\right) \]

   7. hypot-define 68.9%

      \[\leadsto \left(x - y\right) \cdot \left(\frac{{\left(\sqrt[3]{x + y}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{x \cdot x + y \cdot y}}\right) \]

   8. hypot-define 97.7%

      \[\leadsto \left(x - y\right) \cdot \left(\frac{{\left(\sqrt[3]{x + y}\right)}^{2}}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{\sqrt[3]{x + y}}{\color{blue}{\mathsf{hypot}\left(x, y\right)}}\right) \]

6. Applied egg-rr 97.7%

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

   1. associate-*l/ 97.7%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{x + y}\right)}^{2} \cdot \frac{\sqrt[3]{x + y}}{\mathsf{hypot}\left(x, y\right)}}{\mathsf{hypot}\left(x, y\right)}} \]

   2. div-inv 97.7%

      \[\leadsto \left(x - y\right) \cdot \frac{{\left(\sqrt[3]{x + y}\right)}^{2} \cdot \color{blue}{\left(\sqrt[3]{x + y} \cdot \frac{1}{\mathsf{hypot}\left(x, y\right)}\right)}}{\mathsf{hypot}\left(x, y\right)} \]

   3. associate-*r* 97.7%

      \[\leadsto \left(x - y\right) \cdot \frac{\color{blue}{\left({\left(\sqrt[3]{x + y}\right)}^{2} \cdot \sqrt[3]{x + y}\right) \cdot \frac{1}{\mathsf{hypot}\left(x, y\right)}}}{\mathsf{hypot}\left(x, y\right)} \]

   4. unpow2 97.7%

      \[\leadsto \left(x - y\right) \cdot \frac{\left(\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right)} \cdot \sqrt[3]{x + y}\right) \cdot \frac{1}{\mathsf{hypot}\left(x, y\right)}}{\mathsf{hypot}\left(x, y\right)} \]

   5. add-cube-cbrt 99.7%

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

   6. div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 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 70.0%

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

   1. add-sqr-sqrt 70.0%

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

   2. times-frac 70.4%

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

   3. hypot-define 70.4%

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

   4. hypot-define 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{x}{y}}{\frac{y}{x}}, -1\right)\\ \end{array} \end{array} \]

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) (/ y x)) -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) / (y / x)), -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(y / x)), -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[(y / x), $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 99.6%

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

   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. Add Preprocessing
   3. 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-define 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-define 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 53.9%

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

      1. fma-neg 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. times-frac 82.4%

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

      5. unpow2 82.4%

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

      6. metadata-eval 82.4%

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

   9. Simplified 82.4%

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

       1. unpow2 82.4%

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

       2. clear-num 82.4%

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

       3. un-div-inv 82.4%

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

   11. Applied egg-rr 82.4%

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

4. Final simplification 94.5%

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

Alternative 5: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{y}{\left(y + \frac{x}{y} \cdot \left(y - x\right)\right) - x}}\\ \end{array} \end{array} \]

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 (/ y (- (+ y (* (/ x y) (- y x))) x))))))

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 / (y / ((y + ((x / y) * (y - x))) - x));
	}
	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) / (y / ((y + ((x / y) * (y - x))) - x))
    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 / (y / ((y + ((x / y) * (y - x))) - x));
	}
	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 / (y / ((y + ((x / y) * (y - x))) - x))
	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(y / Float64(Float64(y + Float64(Float64(x / y) * Float64(y - x))) - x)));
	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 / (y / ((y + ((x / y) * (y - x))) - x));
	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[(y / N[(N[(y + N[(N[(x / y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $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 99.6%

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

   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%

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

      2. +-commutative 3.1%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in y around inf 80.6%

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

      1. associate-*r/ 81.0%

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

      2. clear-num 81.0%

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

   7. Applied egg-rr 81.0%

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

      1. distribute-rgt-in 81.0%

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

      2. *-un-lft-identity 81.0%

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

      3. associate-+l- 81.0%

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

   9. Applied egg-rr 81.0%

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

4. Final simplification 94.1%

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

Alternative 6: 43.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-169}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{y}{\left(y + \frac{x}{y} \cdot \left(y - x\right)\right) - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-169)
   (- 1.0 (/ (* 2.0 (* y (/ y x))) x))
   (/ -1.0 (/ y (- (+ y (* (/ x y) (- y x))) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-169) {
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	} else {
		tmp = -1.0 / (y / ((y + ((x / y) * (y - x))) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-169) then
        tmp = 1.0d0 - ((2.0d0 * (y * (y / x))) / x)
    else
        tmp = (-1.0d0) / (y / ((y + ((x / y) * (y - x))) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-169) {
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	} else {
		tmp = -1.0 / (y / ((y + ((x / y) * (y - x))) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-169:
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x)
	else:
		tmp = -1.0 / (y / ((y + ((x / y) * (y - x))) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-169)
		tmp = Float64(1.0 - Float64(Float64(2.0 * Float64(y * Float64(y / x))) / x));
	else
		tmp = Float64(-1.0 / Float64(y / Float64(Float64(y + Float64(Float64(x / y) * Float64(y - x))) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-169)
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	else
		tmp = -1.0 / (y / ((y + ((x / y) * (y - x))) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-169], N[(1.0 - N[(N[(2.0 * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(y / N[(N[(y + N[(N[(x / y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1000000000000002e-169

   1. Initial program 63.5%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. fma-define 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around -inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. unsub-neg 37.7%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in y around 0 37.7%

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

      1. div-inv 37.7%

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

      2. unpow2 37.7%

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

      3. associate-*r* 38.6%

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

      4. div-inv 38.6%

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

   10. Applied egg-rr 38.6%

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

   if 3.1000000000000002e-169 < y

   1. Initial program 94.4%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 76.2%

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

      1. associate-*r/ 76.3%

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

      2. clear-num 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. distribute-rgt-in 76.4%

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

      2. *-un-lft-identity 76.4%

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

      3. associate-+l- 76.4%

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

   9. Applied egg-rr 76.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-169}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{y}{\left(y + \frac{x}{y} \cdot \left(y - x\right)\right) - x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-169) 1.0 (* (+ (/ x y) 1.0) (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-169) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) + 1.0) * ((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 <= 2.2d-169) then
        tmp = 1.0d0
    else
        tmp = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-169) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-169:
		tmp = 1.0
	else:
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-169)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x / y) + 1.0) * Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-169)
		tmp = 1.0;
	else
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-169], 1.0, N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.20000000000000007e-169

   1. Initial program 63.5%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. fma-define 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around inf 36.4%

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

   if 2.20000000000000007e-169 < y

   1. Initial program 94.4%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 76.2%

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

      1. associate-*r/ 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-/l* 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.3%

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

       2. *-commutative 76.3%

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

       3. associate-*r/ 76.4%

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

       4. div-sub 76.3%

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

       5. sub-neg 76.3%

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

       6. *-inverses 76.3%

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

       7. metadata-eval 76.3%

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

   11. Simplified 76.3%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-169}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{y}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-169)
   (* (- x y) (/ (+ (/ y x) 1.0) x))
   (* (+ (/ x y) 1.0) (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-169) {
		tmp = (x - y) * (((y / x) + 1.0) / x);
	} else {
		tmp = ((x / y) + 1.0) * ((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 <= 2.5d-169) then
        tmp = (x - y) * (((y / x) + 1.0d0) / x)
    else
        tmp = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-169) {
		tmp = (x - y) * (((y / x) + 1.0) / x);
	} else {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e-169:
		tmp = (x - y) * (((y / x) + 1.0) / x)
	else:
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-169)
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(y / x) + 1.0) / x));
	else
		tmp = Float64(Float64(Float64(x / y) + 1.0) * Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-169)
		tmp = (x - y) * (((y / x) + 1.0) / x);
	else
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-169], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5000000000000001e-169

   1. Initial program 63.5%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. fma-define 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around inf 37.8%

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

   if 2.5000000000000001e-169 < y

   1. Initial program 94.4%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 76.2%

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

      1. associate-*r/ 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-/l* 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.3%

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

       2. *-commutative 76.3%

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

       3. associate-*r/ 76.4%

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

       4. div-sub 76.3%

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

       5. sub-neg 76.3%

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

       6. *-inverses 76.3%

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

       7. metadata-eval 76.3%

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

   11. Simplified 76.3%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-169}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{y}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-169}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e-169)
   (- 1.0 (/ (* 2.0 (* y (/ y x))) x))
   (* (+ (/ x y) 1.0) (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-169) {
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	} else {
		tmp = ((x / y) + 1.0) * ((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 <= 1.2d-169) then
        tmp = 1.0d0 - ((2.0d0 * (y * (y / x))) / x)
    else
        tmp = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-169) {
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	} else {
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e-169:
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x)
	else:
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-169)
		tmp = Float64(1.0 - Float64(Float64(2.0 * Float64(y * Float64(y / x))) / x));
	else
		tmp = Float64(Float64(Float64(x / y) + 1.0) * Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e-169)
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	else
		tmp = ((x / y) + 1.0) * ((x / y) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e-169], N[(1.0 - N[(N[(2.0 * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.20000000000000005e-169

   1. Initial program 63.5%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. fma-define 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around -inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. unsub-neg 37.7%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in y around 0 37.7%

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

      1. div-inv 37.7%

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

      2. unpow2 37.7%

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

      3. associate-*r* 38.6%

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

      4. div-inv 38.6%

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

   10. Applied egg-rr 38.6%

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

   if 1.20000000000000005e-169 < y

   1. Initial program 94.4%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 76.2%

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

      1. associate-*r/ 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-/l* 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.3%

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

       2. *-commutative 76.3%

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

       3. associate-*r/ 76.4%

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

       4. div-sub 76.3%

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

       5. sub-neg 76.3%

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

       6. *-inverses 76.3%

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

       7. metadata-eval 76.3%

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

   11. Simplified 76.3%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-169}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-170}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(\frac{x}{y} + 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e-170)
   (- 1.0 (/ (* 2.0 (* y (/ y x))) x))
   (/ (* (- x y) (+ (/ x y) 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-170) {
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	} else {
		tmp = ((x - y) * ((x / y) + 1.0)) / 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 <= 2d-170) then
        tmp = 1.0d0 - ((2.0d0 * (y * (y / x))) / x)
    else
        tmp = ((x - y) * ((x / y) + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-170) {
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	} else {
		tmp = ((x - y) * ((x / y) + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-170:
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x)
	else:
		tmp = ((x - y) * ((x / y) + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-170)
		tmp = Float64(1.0 - Float64(Float64(2.0 * Float64(y * Float64(y / x))) / x));
	else
		tmp = Float64(Float64(Float64(x - y) * Float64(Float64(x / y) + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-170)
		tmp = 1.0 - ((2.0 * (y * (y / x))) / x);
	else
		tmp = ((x - y) * ((x / y) + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-170], N[(1.0 - N[(N[(2.0 * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999997e-170

   1. Initial program 63.5%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. fma-define 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around -inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. unsub-neg 37.7%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in y around 0 37.7%

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

      1. div-inv 37.7%

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

      2. unpow2 37.7%

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

      3. associate-*r* 38.6%

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

      4. div-inv 38.6%

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

   10. Applied egg-rr 38.6%

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

   if 1.99999999999999997e-170 < y

   1. Initial program 94.4%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 76.2%

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

      1. associate-*r/ 76.3%

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

   7. Applied egg-rr 76.3%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-170}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(\frac{x}{y} + 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-169) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-169) {
		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 <= 2d-169) 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 <= 2e-169) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-169:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-169)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-169)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-169], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000004e-169

   1. Initial program 63.5%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. fma-define 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around inf 36.4%

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

   if 2.00000000000000004e-169 < y

   1. Initial program 94.4%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in x around 0 75.2%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.2% 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 70.0%

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

   1. associate-/l* 70.2%

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

   2. +-commutative 70.2%

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

   3. fma-define 70.2%

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

3. Simplified 70.2%

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

5. Taylor expanded in x around 0 65.7%

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

6. Final simplification 65.7%

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

Developer target: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t\_0 \land t\_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024077 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))