Kahan p9 Example

Percentage Accurate: 68.6% → 100.0%

Time: 10.1s

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: 68.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. fma-define 68.7%

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

   2. add-sqr-sqrt 68.7%

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

   3. times-frac 68.8%

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

   4. fma-define 68.8%

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

   5. hypot-define 68.8%

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

   6. fma-define 68.8%

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

   7. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. 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. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 92.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   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. fma-define 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 3.1%

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

      4. fma-define 3.1%

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

      5. hypot-define 3.1%

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

      6. fma-define 3.1%

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

      7. 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. Taylor expanded in x around 0 52.5%

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

      1. fma-neg 52.5%

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

      2. unpow2 52.5%

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

      3. unpow2 52.5%

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

      4. times-frac 80.2%

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

      5. unpow2 80.2%

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

      6. metadata-eval 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \]
5. 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 68.7%

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

   1. fma-define 68.7%

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

   2. add-sqr-sqrt 68.7%

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

   3. times-frac 68.8%

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

   4. fma-define 68.8%

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

   5. hypot-define 68.8%

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

   6. fma-define 68.8%

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

   7. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 72.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (* (/ (- x y) (hypot 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 = ((x - y) / hypot(x, y)) * ((x / y) + 1.0);
	}
	return tmp;
}

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) / Math.hypot(x, y)) * ((x / y) + 1.0);
	}
	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) / math.hypot(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 = Float64(Float64(Float64(x - y) / hypot(x, y)) * Float64(Float64(x / y) + 1.0));
	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) / hypot(x, y)) * ((x / y) + 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]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + 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%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   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. fma-define 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 3.1%

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

      4. fma-define 3.1%

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

      5. hypot-define 3.1%

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

      6. fma-define 3.1%

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

      7. 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. Taylor expanded in x around 0 14.1%

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

      1. +-commutative 14.1%

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

   7. Simplified 14.1%

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

4. Final simplification 73.1%

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

Alternative 5: 92.0% accurate, 0.4× speedup

Math

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

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

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   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. fma-define 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 78.2%

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

      1. un-div-inv 78.4%

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

   7. Applied egg-rr 78.4%

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

4. Final simplification 93.2%

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

Alternative 6: 41.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{x}\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-149}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) x)))
   (if (<= y 5.2e-198)
     t_0
     (if (<= y 3.9e-149)
       (/ (- x y) y)
       (if (<= y 1.4e-134) t_0 (* (- x y) (/ (+ (/ x y) 1.0) y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / x;
	double tmp;
	if (y <= 5.2e-198) {
		tmp = t_0;
	} else if (y <= 3.9e-149) {
		tmp = (x - y) / y;
	} else if (y <= 1.4e-134) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / x
    if (y <= 5.2d-198) then
        tmp = t_0
    else if (y <= 3.9d-149) then
        tmp = (x - y) / y
    else if (y <= 1.4d-134) then
        tmp = t_0
    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 t_0 = (x - y) / x;
	double tmp;
	if (y <= 5.2e-198) {
		tmp = t_0;
	} else if (y <= 3.9e-149) {
		tmp = (x - y) / y;
	} else if (y <= 1.4e-134) {
		tmp = t_0;
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / x
	tmp = 0
	if y <= 5.2e-198:
		tmp = t_0
	elif y <= 3.9e-149:
		tmp = (x - y) / y
	elif y <= 1.4e-134:
		tmp = t_0
	else:
		tmp = (x - y) * (((x / y) + 1.0) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / x)
	tmp = 0.0
	if (y <= 5.2e-198)
		tmp = t_0;
	elseif (y <= 3.9e-149)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 1.4e-134)
		tmp = t_0;
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / x;
	tmp = 0.0;
	if (y <= 5.2e-198)
		tmp = t_0;
	elseif (y <= 3.9e-149)
		tmp = (x - y) / y;
	elseif (y <= 1.4e-134)
		tmp = t_0;
	else
		tmp = (x - y) * (((x / y) + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, 5.2e-198], t$95$0, If[LessEqual[y, 3.9e-149], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.4e-134], t$95$0, N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.20000000000000014e-198 or 3.9000000000000002e-149 < y < 1.3999999999999999e-134

   1. Initial program 65.2%

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

      1. associate-/l* 65.0%

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

      2. fma-define 65.0%

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

   3. Simplified 65.0%

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

   5. Taylor expanded in x around inf 33.4%

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

      1. un-div-inv 33.5%

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

   7. Applied egg-rr 33.5%

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

   if 5.20000000000000014e-198 < y < 3.9000000000000002e-149

   1. Initial program 27.3%

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

      1. associate-/l* 27.3%

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

      2. fma-define 27.3%

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

   3. Simplified 27.3%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. un-div-inv 68.3%

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

   7. Applied egg-rr 68.3%

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

   if 1.3999999999999999e-134 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.6%

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

      2. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 82.8%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-149}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-198}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-148}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-198)
   (* (- x y) (/ (+ 1.0 (/ y x)) x))
   (if (<= y 1.05e-148)
     (/ (- x y) y)
     (if (<= y 7.8e-132) (/ (- x y) x) (* (- x y) (/ (+ (/ x y) 1.0) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-198) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else if (y <= 1.05e-148) {
		tmp = (x - y) / y;
	} else if (y <= 7.8e-132) {
		tmp = (x - y) / 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 <= 4.8d-198) then
        tmp = (x - y) * ((1.0d0 + (y / x)) / x)
    else if (y <= 1.05d-148) then
        tmp = (x - y) / y
    else if (y <= 7.8d-132) then
        tmp = (x - y) / 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 <= 4.8e-198) {
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	} else if (y <= 1.05e-148) {
		tmp = (x - y) / y;
	} else if (y <= 7.8e-132) {
		tmp = (x - y) / x;
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e-198:
		tmp = (x - y) * ((1.0 + (y / x)) / x)
	elif y <= 1.05e-148:
		tmp = (x - y) / y
	elif y <= 7.8e-132:
		tmp = (x - y) / x
	else:
		tmp = (x - y) * (((x / y) + 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-198)
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 + Float64(y / x)) / x));
	elseif (y <= 1.05e-148)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 7.8e-132)
		tmp = Float64(Float64(x - y) / x);
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-198)
		tmp = (x - y) * ((1.0 + (y / x)) / x);
	elseif (y <= 1.05e-148)
		tmp = (x - y) / y;
	elseif (y <= 7.8e-132)
		tmp = (x - y) / x;
	else
		tmp = (x - y) * (((x / y) + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-198], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e-148], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 7.8e-132], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.79999999999999973e-198

   1. Initial program 64.2%

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

      1. associate-/l* 63.9%

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

      2. fma-define 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around inf 34.4%

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

   if 4.79999999999999973e-198 < y < 1.05e-148

   1. Initial program 27.3%

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

      1. associate-/l* 27.3%

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

      2. fma-define 27.3%

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

   3. Simplified 27.3%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. un-div-inv 68.3%

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

   7. Applied egg-rr 68.3%

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

   if 1.05e-148 < y < 7.79999999999999964e-132

   1. Initial program 99.5%

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

      1. associate-/l* 99.7%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 72.9%

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

      1. un-div-inv 72.9%

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

   7. Applied egg-rr 72.9%

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

   if 7.79999999999999964e-132 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.6%

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

      2. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 82.8%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1 + \frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-198}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-148}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;1 + -1.5 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-149}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-198)
   (+ 1.0 (* -1.5 (/ (/ y x) (/ x y))))
   (if (<= y 5.6e-149)
     (/ (- x y) y)
     (if (<= y 3.7e-134) (/ (- x y) x) (* (- x y) (/ (+ (/ x y) 1.0) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-198) {
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	} else if (y <= 5.6e-149) {
		tmp = (x - y) / y;
	} else if (y <= 3.7e-134) {
		tmp = (x - y) / 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 <= 5.2d-198) then
        tmp = 1.0d0 + ((-1.5d0) * ((y / x) / (x / y)))
    else if (y <= 5.6d-149) then
        tmp = (x - y) / y
    else if (y <= 3.7d-134) then
        tmp = (x - y) / 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 <= 5.2e-198) {
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	} else if (y <= 5.6e-149) {
		tmp = (x - y) / y;
	} else if (y <= 3.7e-134) {
		tmp = (x - y) / x;
	} else {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-198:
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)))
	elif y <= 5.6e-149:
		tmp = (x - y) / y
	elif y <= 3.7e-134:
		tmp = (x - y) / x
	else:
		tmp = (x - y) * (((x / y) + 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-198)
		tmp = Float64(1.0 + Float64(-1.5 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (y <= 5.6e-149)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 3.7e-134)
		tmp = Float64(Float64(x - y) / x);
	else
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-198)
		tmp = 1.0 + (-1.5 * ((y / x) / (x / y)));
	elseif (y <= 5.6e-149)
		tmp = (x - y) / y;
	elseif (y <= 3.7e-134)
		tmp = (x - y) / x;
	else
		tmp = (x - y) * (((x / y) + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e-198], N[(1.0 + N[(-1.5 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e-149], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 3.7e-134], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.20000000000000014e-198

   1. Initial program 64.2%

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

      1. fma-define 64.2%

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

      2. add-sqr-sqrt 64.2%

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

      3. times-frac 64.2%

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

      4. fma-define 64.2%

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

      5. hypot-define 64.3%

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

      6. fma-define 64.3%

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

      7. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 35.1%

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

   6. Taylor expanded in x around inf 26.3%

      \[\leadsto \color{blue}{1 + \left(-1 \cdot \frac{y}{x} + \left(-1 \cdot \frac{{y}^{2}}{{x}^{2}} + \left(-0.5 \cdot \frac{{y}^{2}}{{x}^{2}} + \frac{y}{x}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 26.3%

         \[\leadsto 1 + \color{blue}{\left(\left(-1 \cdot \frac{{y}^{2}}{{x}^{2}} + \left(-0.5 \cdot \frac{{y}^{2}}{{x}^{2}} + \frac{y}{x}\right)\right) + -1 \cdot \frac{y}{x}\right)} \]

      2. associate-+r+ 26.3%

         \[\leadsto 1 + \left(\color{blue}{\left(\left(-1 \cdot \frac{{y}^{2}}{{x}^{2}} + -0.5 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + \frac{y}{x}\right)} + -1 \cdot \frac{y}{x}\right) \]

      3. distribute-rgt-out 26.3%

         \[\leadsto 1 + \left(\left(\color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot \left(-1 + -0.5\right)} + \frac{y}{x}\right) + -1 \cdot \frac{y}{x}\right) \]

      4. metadata-eval 26.3%

         \[\leadsto 1 + \left(\left(\frac{{y}^{2}}{{x}^{2}} \cdot \color{blue}{-1.5} + \frac{y}{x}\right) + -1 \cdot \frac{y}{x}\right) \]

      5. *-commutative 26.3%

         \[\leadsto 1 + \left(\left(\color{blue}{-1.5 \cdot \frac{{y}^{2}}{{x}^{2}}} + \frac{y}{x}\right) + -1 \cdot \frac{y}{x}\right) \]

      6. associate-+l+ 26.3%

         \[\leadsto 1 + \color{blue}{\left(-1.5 \cdot \frac{{y}^{2}}{{x}^{2}} + \left(\frac{y}{x} + -1 \cdot \frac{y}{x}\right)\right)} \]

      7. unpow2 26.3%

         \[\leadsto 1 + \left(-1.5 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} + \left(\frac{y}{x} + -1 \cdot \frac{y}{x}\right)\right) \]

      8. unpow2 26.3%

         \[\leadsto 1 + \left(-1.5 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} + \left(\frac{y}{x} + -1 \cdot \frac{y}{x}\right)\right) \]

      9. times-frac 34.0%

         \[\leadsto 1 + \left(-1.5 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} + \left(\frac{y}{x} + -1 \cdot \frac{y}{x}\right)\right) \]

      10. unpow2 34.0%

          \[\leadsto 1 + \left(-1.5 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} + \left(\frac{y}{x} + -1 \cdot \frac{y}{x}\right)\right) \]

      11. distribute-rgt1-in 34.0%

          \[\leadsto 1 + \left(-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + \color{blue}{\left(-1 + 1\right) \cdot \frac{y}{x}}\right) \]

      12. metadata-eval 34.0%

          \[\leadsto 1 + \left(-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + \color{blue}{0} \cdot \frac{y}{x}\right) \]

      13. mul0-lft 34.6%

          \[\leadsto 1 + \left(-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + \color{blue}{0}\right) \]

   8. Simplified 34.6%

      \[\leadsto \color{blue}{1 + \left(-1.5 \cdot {\left(\frac{y}{x}\right)}^{2} + 0\right)} \]
   9. Step-by-step derivation

      1. unpow2 34.6%

         \[\leadsto 1 + \left(-1.5 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} + 0\right) \]

      2. clear-num 34.6%

         \[\leadsto 1 + \left(-1.5 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) + 0\right) \]

      3. un-div-inv 34.6%

         \[\leadsto 1 + \left(-1.5 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} + 0\right) \]

   10. Applied egg-rr 34.6%

       \[\leadsto 1 + \left(-1.5 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} + 0\right) \]

   if 5.20000000000000014e-198 < y < 5.5999999999999997e-149

   1. Initial program 27.3%

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

      1. associate-/l* 27.3%

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

      2. fma-define 27.3%

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

   3. Simplified 27.3%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. un-div-inv 68.3%

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

   7. Applied egg-rr 68.3%

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

   if 5.5999999999999997e-149 < y < 3.7e-134

   1. Initial program 99.5%

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

      1. associate-/l* 99.7%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 72.9%

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

      1. un-div-inv 72.9%

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

   7. Applied egg-rr 72.9%

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

   if 3.7e-134 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.6%

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

      2. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 82.8%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1 + \frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;1 + -1.5 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-149}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-198} \lor \neg \left(y \leq 6.8 \cdot 10^{-149}\right) \land y \leq 6.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 3e-198) (and (not (<= y 6.8e-149)) (<= y 6.4e-134)))
   (/ (- x y) x)
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 3e-198) || (!(y <= 6.8e-149) && (y <= 6.4e-134))) {
		tmp = (x - y) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 3d-198) .or. (.not. (y <= 6.8d-149)) .and. (y <= 6.4d-134)) then
        tmp = (x - y) / x
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 3e-198) || (!(y <= 6.8e-149) && (y <= 6.4e-134))) {
		tmp = (x - y) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 3e-198) or (not (y <= 6.8e-149) and (y <= 6.4e-134)):
		tmp = (x - y) / x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 3e-198) || (!(y <= 6.8e-149) && (y <= 6.4e-134)))
		tmp = Float64(Float64(x - y) / x);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 3e-198) || (~((y <= 6.8e-149)) && (y <= 6.4e-134)))
		tmp = (x - y) / x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 3e-198], And[N[Not[LessEqual[y, 6.8e-149]], $MachinePrecision], LessEqual[y, 6.4e-134]]], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000001e-198 or 6.7999999999999998e-149 < y < 6.4000000000000003e-134

   1. Initial program 65.2%

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

      1. associate-/l* 65.0%

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

      2. fma-define 65.0%

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

   3. Simplified 65.0%

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

   5. Taylor expanded in x around inf 33.4%

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

      1. un-div-inv 33.5%

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

   7. Applied egg-rr 33.5%

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

   if 3.0000000000000001e-198 < y < 6.7999999999999998e-149 or 6.4000000000000003e-134 < y

   1. Initial program 83.6%

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

      1. associate-/l* 83.4%

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

      2. fma-define 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 78.5%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-198} \lor \neg \left(y \leq 6.8 \cdot 10^{-149}\right) \land y \leq 6.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{x}\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-149}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) x)))
   (if (<= y 5.2e-198)
     t_0
     (if (<= y 4.3e-149) (/ (- x y) y) (if (<= y 6.4e-131) t_0 -1.0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / x;
	double tmp;
	if (y <= 5.2e-198) {
		tmp = t_0;
	} else if (y <= 4.3e-149) {
		tmp = (x - y) / y;
	} else if (y <= 6.4e-131) {
		tmp = t_0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / x
    if (y <= 5.2d-198) then
        tmp = t_0
    else if (y <= 4.3d-149) then
        tmp = (x - y) / y
    else if (y <= 6.4d-131) then
        tmp = t_0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / x;
	double tmp;
	if (y <= 5.2e-198) {
		tmp = t_0;
	} else if (y <= 4.3e-149) {
		tmp = (x - y) / y;
	} else if (y <= 6.4e-131) {
		tmp = t_0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / x
	tmp = 0
	if y <= 5.2e-198:
		tmp = t_0
	elif y <= 4.3e-149:
		tmp = (x - y) / y
	elif y <= 6.4e-131:
		tmp = t_0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / x)
	tmp = 0.0
	if (y <= 5.2e-198)
		tmp = t_0;
	elseif (y <= 4.3e-149)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 6.4e-131)
		tmp = t_0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / x;
	tmp = 0.0;
	if (y <= 5.2e-198)
		tmp = t_0;
	elseif (y <= 4.3e-149)
		tmp = (x - y) / y;
	elseif (y <= 6.4e-131)
		tmp = t_0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, 5.2e-198], t$95$0, If[LessEqual[y, 4.3e-149], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 6.4e-131], t$95$0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.20000000000000014e-198 or 4.30000000000000037e-149 < y < 6.3999999999999999e-131

   1. Initial program 65.2%

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

      1. associate-/l* 65.0%

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

      2. fma-define 65.0%

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

   3. Simplified 65.0%

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

   5. Taylor expanded in x around inf 33.4%

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

      1. un-div-inv 33.5%

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

   7. Applied egg-rr 33.5%

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

   if 5.20000000000000014e-198 < y < 4.30000000000000037e-149

   1. Initial program 27.3%

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

      1. associate-/l* 27.3%

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

      2. fma-define 27.3%

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

   3. Simplified 27.3%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. un-div-inv 68.3%

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

   7. Applied egg-rr 68.3%

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

   if 6.3999999999999999e-131 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.6%

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

      2. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 81.7%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-149}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-131}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.12 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.12e-186)
   1.0
   (if (<= y 7.2e-149) -1.0 (if (<= y 6e-134) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.12e-186) {
		tmp = 1.0;
	} else if (y <= 7.2e-149) {
		tmp = -1.0;
	} else if (y <= 6e-134) {
		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 <= 5.12d-186) then
        tmp = 1.0d0
    else if (y <= 7.2d-149) then
        tmp = -1.0d0
    else if (y <= 6d-134) 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 <= 5.12e-186) {
		tmp = 1.0;
	} else if (y <= 7.2e-149) {
		tmp = -1.0;
	} else if (y <= 6e-134) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.12e-186:
		tmp = 1.0
	elif y <= 7.2e-149:
		tmp = -1.0
	elif y <= 6e-134:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.12e-186)
		tmp = 1.0;
	elseif (y <= 7.2e-149)
		tmp = -1.0;
	elseif (y <= 6e-134)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.12e-186)
		tmp = 1.0;
	elseif (y <= 7.2e-149)
		tmp = -1.0;
	elseif (y <= 6e-134)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.12e-186], 1.0, If[LessEqual[y, 7.2e-149], -1.0, If[LessEqual[y, 6e-134], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1199999999999997e-186 or 7.2000000000000004e-149 < y < 6e-134

   1. Initial program 64.0%

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

      1. associate-/l* 63.8%

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

      2. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 33.9%

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

   if 5.1199999999999997e-186 < y < 7.2000000000000004e-149 or 6e-134 < y

   1. Initial program 91.1%

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

      1. associate-/l* 90.5%

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

      2. fma-define 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around 0 78.8%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.12 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% 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 68.7%

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

   1. associate-/l* 68.5%

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

   2. fma-define 68.5%

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

3. Simplified 68.5%

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

5. Taylor expanded in x around 0 68.2%

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

6. Final simplification 68.2%

   \[\leadsto -1 \]
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 2024066 
(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))))