Kahan p9 Example

Percentage Accurate: 67.8% → 99.9%

Time: 8.7s

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: 67.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. add-sqr-sqrt 69.9%

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

   2. times-frac 70.2%

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

   3. hypot-define 70.3%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 92.7% 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. *-commutative 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. fma-neg 55.8%

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

      2. unpow2 55.8%

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

      3. unpow2 55.8%

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

      4. times-frac 79.7%

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

      5. unpow2 79.7%

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

      6. metadata-eval 79.7%

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

   7. Simplified 79.7%

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

Alternative 3: 72.2% 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. 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 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 0 14.0%

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

4. Final simplification 74.1%

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

Alternative 4: 92.5% accurate, 0.4× speedup

Math

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

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) 1.0) (+ (/ 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) + 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) :: t_0
    real(8) :: tmp
    t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = ((x / y) + 1.0d0) * ((x / y) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((x / y) + 1.0) * ((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) + 1.0) * ((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) + 1.0) * 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) + 1.0) * ((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] + 1.0), $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. 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 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 0 14.0%

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

   6. Taylor expanded in x around 0 79.2%

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

4. Final simplification 93.7%

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

Alternative 5: 41.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-154}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-132} \lor \neg \left(y \leq 7.5 \cdot 10^{-124}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x} - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.65e-154)
   (- 1.0 (/ y x))
   (if (or (<= y 1.16e-132) (not (<= y 7.5e-124)))
     (* (- x y) (/ (+ (/ x y) 1.0) y))
     (- (/ x x) (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-154) {
		tmp = 1.0 - (y / x);
	} else if ((y <= 1.16e-132) || !(y <= 7.5e-124)) {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	} else {
		tmp = (x / x) - (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.65d-154) then
        tmp = 1.0d0 - (y / x)
    else if ((y <= 1.16d-132) .or. (.not. (y <= 7.5d-124))) then
        tmp = (x - y) * (((x / y) + 1.0d0) / y)
    else
        tmp = (x / x) - (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-154) {
		tmp = 1.0 - (y / x);
	} else if ((y <= 1.16e-132) || !(y <= 7.5e-124)) {
		tmp = (x - y) * (((x / y) + 1.0) / y);
	} else {
		tmp = (x / x) - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.65e-154:
		tmp = 1.0 - (y / x)
	elif (y <= 1.16e-132) or not (y <= 7.5e-124):
		tmp = (x - y) * (((x / y) + 1.0) / y)
	else:
		tmp = (x / x) - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.65e-154)
		tmp = Float64(1.0 - Float64(y / x));
	elseif ((y <= 1.16e-132) || !(y <= 7.5e-124))
		tmp = Float64(Float64(x - y) * Float64(Float64(Float64(x / y) + 1.0) / y));
	else
		tmp = Float64(Float64(x / x) - Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.65e-154)
		tmp = 1.0 - (y / x);
	elseif ((y <= 1.16e-132) || ~((y <= 7.5e-124)))
		tmp = (x - y) * (((x / y) + 1.0) / y);
	else
		tmp = (x / x) - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.65e-154], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.16e-132], N[Not[LessEqual[y, 7.5e-124]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / x), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.65000000000000014e-154

   1. Initial program 63.6%

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

      1. associate-/l* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

   8. Taylor expanded in x around inf 32.4%

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

      1. neg-mul-1 32.4%

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

      2. sub-neg 32.4%

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

   10. Simplified 32.4%

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

   if 1.65000000000000014e-154 < y < 1.1599999999999999e-132 or 7.4999999999999996e-124 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 79.6%

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

   if 1.1599999999999999e-132 < y < 7.4999999999999996e-124

   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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. un-div-inv 100.0%

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

      2. div-sub 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-154}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-132} \lor \neg \left(y \leq 7.5 \cdot 10^{-124}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x} - \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;y \leq 9 \cdot 10^{-155}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-132}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_0}{y}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{x} - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (<= y 9e-155)
     (- 1.0 (/ y x))
     (if (<= y 5.8e-132)
       (* (- x y) (/ t_0 y))
       (if (<= y 6.9e-124) (- (/ x x) (/ y x)) (* t_0 (+ (/ x y) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (y <= 9e-155) {
		tmp = 1.0 - (y / x);
	} else if (y <= 5.8e-132) {
		tmp = (x - y) * (t_0 / y);
	} else if (y <= 6.9e-124) {
		tmp = (x / x) - (y / x);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (x / y) + 1.0d0
    if (y <= 9d-155) then
        tmp = 1.0d0 - (y / x)
    else if (y <= 5.8d-132) then
        tmp = (x - y) * (t_0 / y)
    else if (y <= 6.9d-124) then
        tmp = (x / x) - (y / x)
    else
        tmp = t_0 * ((x / y) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (y <= 9e-155) {
		tmp = 1.0 - (y / x);
	} else if (y <= 5.8e-132) {
		tmp = (x - y) * (t_0 / y);
	} else if (y <= 6.9e-124) {
		tmp = (x / x) - (y / x);
	} else {
		tmp = t_0 * ((x / y) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if y <= 9e-155:
		tmp = 1.0 - (y / x)
	elif y <= 5.8e-132:
		tmp = (x - y) * (t_0 / y)
	elif y <= 6.9e-124:
		tmp = (x / x) - (y / x)
	else:
		tmp = t_0 * ((x / y) + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if (y <= 9e-155)
		tmp = Float64(1.0 - Float64(y / x));
	elseif (y <= 5.8e-132)
		tmp = Float64(Float64(x - y) * Float64(t_0 / y));
	elseif (y <= 6.9e-124)
		tmp = Float64(Float64(x / x) - Float64(y / x));
	else
		tmp = Float64(t_0 * Float64(Float64(x / y) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if (y <= 9e-155)
		tmp = 1.0 - (y / x);
	elseif (y <= 5.8e-132)
		tmp = (x - y) * (t_0 / y);
	elseif (y <= 6.9e-124)
		tmp = (x / x) - (y / x);
	else
		tmp = t_0 * ((x / y) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 9e-155], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-132], N[(N[(x - y), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.9e-124], N[(N[(x / x), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 9.0000000000000007e-155

   1. Initial program 63.6%

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

      1. associate-/l* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

   8. Taylor expanded in x around inf 32.4%

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

      1. neg-mul-1 32.4%

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

      2. sub-neg 32.4%

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

   10. Simplified 32.4%

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

   if 9.0000000000000007e-155 < y < 5.79999999999999967e-132

   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* 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 62.7%

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

   if 5.79999999999999967e-132 < y < 6.9e-124

   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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. un-div-inv 100.0%

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

      2. div-sub 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 6.9e-124 < y

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 99.9%

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

      2. times-frac 99.8%

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

      3. hypot-define 99.8%

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

      4. hypot-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 82.6%

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

   6. Taylor expanded in x around 0 82.2%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-155}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-132}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{x} - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.14 \cdot 10^{-154}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-133}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{x} - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.14e-154)
   (- 1.0 (/ y x))
   (if (<= y 7.4e-133) -1.0 (if (<= y 6.9e-124) (- (/ x x) (/ y x)) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.14e-154) {
		tmp = 1.0 - (y / x);
	} else if (y <= 7.4e-133) {
		tmp = -1.0;
	} else if (y <= 6.9e-124) {
		tmp = (x / 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 <= 1.14d-154) then
        tmp = 1.0d0 - (y / x)
    else if (y <= 7.4d-133) then
        tmp = -1.0d0
    else if (y <= 6.9d-124) then
        tmp = (x / 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 <= 1.14e-154) {
		tmp = 1.0 - (y / x);
	} else if (y <= 7.4e-133) {
		tmp = -1.0;
	} else if (y <= 6.9e-124) {
		tmp = (x / x) - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.14e-154:
		tmp = 1.0 - (y / x)
	elif y <= 7.4e-133:
		tmp = -1.0
	elif y <= 6.9e-124:
		tmp = (x / x) - (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.14e-154)
		tmp = Float64(1.0 - Float64(y / x));
	elseif (y <= 7.4e-133)
		tmp = -1.0;
	elseif (y <= 6.9e-124)
		tmp = Float64(Float64(x / x) - Float64(y / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.14e-154)
		tmp = 1.0 - (y / x);
	elseif (y <= 7.4e-133)
		tmp = -1.0;
	elseif (y <= 6.9e-124)
		tmp = (x / x) - (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.14e-154], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e-133], -1.0, If[LessEqual[y, 6.9e-124], N[(N[(x / x), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.13999999999999993e-154

   1. Initial program 63.6%

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

      1. associate-/l* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

   8. Taylor expanded in x around inf 32.4%

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

      1. neg-mul-1 32.4%

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

      2. sub-neg 32.4%

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

   10. Simplified 32.4%

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

   if 1.13999999999999993e-154 < y < 7.40000000000000071e-133 or 6.9e-124 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 78.8%

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

   if 7.40000000000000071e-133 < y < 6.9e-124

   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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. un-div-inv 100.0%

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

      2. div-sub 100.0%

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

   7. Applied egg-rr 100.0%

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

Alternative 8: 41.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-154} \lor \neg \left(y \leq 2.25 \cdot 10^{-132}\right) \land y \leq 7.3 \cdot 10^{-124}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.65e-154) (and (not (<= y 2.25e-132)) (<= y 7.3e-124)))
   (- 1.0 (/ y x))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1.65e-154) || (!(y <= 2.25e-132) && (y <= 7.3e-124))) {
		tmp = 1.0 - (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 <= 1.65d-154) .or. (.not. (y <= 2.25d-132)) .and. (y <= 7.3d-124)) then
        tmp = 1.0d0 - (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 <= 1.65e-154) || (!(y <= 2.25e-132) && (y <= 7.3e-124))) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1.65e-154) or (not (y <= 2.25e-132) and (y <= 7.3e-124)):
		tmp = 1.0 - (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 1.65e-154) || (!(y <= 2.25e-132) && (y <= 7.3e-124)))
		tmp = Float64(1.0 - Float64(y / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 1.65e-154) || (~((y <= 2.25e-132)) && (y <= 7.3e-124)))
		tmp = 1.0 - (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.65e-154], And[N[Not[LessEqual[y, 2.25e-132]], $MachinePrecision], LessEqual[y, 7.3e-124]]], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 3 regimes

2. if y < 1.65000000000000014e-154

   1. Initial program 63.6%

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

      1. associate-/l* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

   8. Taylor expanded in x around inf 32.4%

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

      1. neg-mul-1 32.4%

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

      2. sub-neg 32.4%

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

   10. Simplified 32.4%

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

   if 1.65000000000000014e-154 < y < 2.25e-132 or 7.3e-124 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 78.8%

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

   if 2.25e-132 < y < 7.3e-124

   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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. un-div-inv 100.0%

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

      2. div-sub 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

      2. *-inverses 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-154} \lor \neg \left(y \leq 2.25 \cdot 10^{-132}\right) \land y \leq 7.3 \cdot 10^{-124}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-154} \lor \neg \left(y \leq 6.7 \cdot 10^{-133}\right) \land y \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.55e-154) (and (not (<= y 6.7e-133)) (<= y 6.9e-124)))
   (- 1.0 (/ y x))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1.55e-154) || (!(y <= 6.7e-133) && (y <= 6.9e-124))) {
		tmp = 1.0 - (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 <= 1.55d-154) .or. (.not. (y <= 6.7d-133)) .and. (y <= 6.9d-124)) then
        tmp = 1.0d0 - (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 <= 1.55e-154) || (!(y <= 6.7e-133) && (y <= 6.9e-124))) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1.55e-154) or (not (y <= 6.7e-133) and (y <= 6.9e-124)):
		tmp = 1.0 - (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 1.55e-154) || (!(y <= 6.7e-133) && (y <= 6.9e-124)))
		tmp = Float64(1.0 - Float64(y / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 1.55e-154) || (~((y <= 6.7e-133)) && (y <= 6.9e-124)))
		tmp = 1.0 - (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.55e-154], And[N[Not[LessEqual[y, 6.7e-133]], $MachinePrecision], LessEqual[y, 6.9e-124]]], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.54999999999999991e-154 or 6.7000000000000002e-133 < y < 6.9e-124

   1. Initial program 64.7%

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

      1. associate-/l* 64.8%

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

      2. +-commutative 64.8%

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

      3. fma-define 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in x around inf 34.2%

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

      1. un-div-inv 34.3%

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

   7. Applied egg-rr 34.3%

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

   8. Taylor expanded in x around inf 34.3%

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

      1. neg-mul-1 34.3%

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

      2. sub-neg 34.3%

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

   10. Simplified 34.3%

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

   if 1.54999999999999991e-154 < y < 6.7000000000000002e-133 or 6.9e-124 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 78.8%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-154} \lor \neg \left(y \leq 6.7 \cdot 10^{-133}\right) \land y \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.14 \cdot 10^{-154}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-132}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.14e-154)
   (- 1.0 (/ y x))
   (if (<= y 1.6e-132) -1.0 (if (<= y 6.9e-124) (/ (- x y) x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.14e-154) {
		tmp = 1.0 - (y / x);
	} else if (y <= 1.6e-132) {
		tmp = -1.0;
	} else if (y <= 6.9e-124) {
		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 <= 1.14d-154) then
        tmp = 1.0d0 - (y / x)
    else if (y <= 1.6d-132) then
        tmp = -1.0d0
    else if (y <= 6.9d-124) 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 <= 1.14e-154) {
		tmp = 1.0 - (y / x);
	} else if (y <= 1.6e-132) {
		tmp = -1.0;
	} else if (y <= 6.9e-124) {
		tmp = (x - y) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.14e-154:
		tmp = 1.0 - (y / x)
	elif y <= 1.6e-132:
		tmp = -1.0
	elif y <= 6.9e-124:
		tmp = (x - y) / x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.14e-154)
		tmp = Float64(1.0 - Float64(y / x));
	elseif (y <= 1.6e-132)
		tmp = -1.0;
	elseif (y <= 6.9e-124)
		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 <= 1.14e-154)
		tmp = 1.0 - (y / x);
	elseif (y <= 1.6e-132)
		tmp = -1.0;
	elseif (y <= 6.9e-124)
		tmp = (x - y) / x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.14e-154], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-132], -1.0, If[LessEqual[y, 6.9e-124], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.13999999999999993e-154

   1. Initial program 63.6%

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

      1. associate-/l* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

   8. Taylor expanded in x around inf 32.4%

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

      1. neg-mul-1 32.4%

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

      2. sub-neg 32.4%

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

   10. Simplified 32.4%

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

   if 1.13999999999999993e-154 < y < 1.6000000000000001e-132 or 6.9e-124 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 78.8%

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

   if 1.6000000000000001e-132 < y < 6.9e-124

   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* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

Alternative 11: 41.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.2e-163) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-163) {
		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 <= 3.2d-163) 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 <= 3.2e-163) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-163:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-163)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-163)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-163], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.19999999999999988e-163

   1. Initial program 63.1%

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

      1. associate-/l* 63.7%

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

      2. +-commutative 63.7%

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

      3. fma-define 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in x around inf 32.7%

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

   if 3.19999999999999988e-163 < 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* 97.9%

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

      2. +-commutative 97.9%

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

      3. fma-define 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 68.2%

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

Alternative 12: 66.1% 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 69.9%

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

   1. associate-/l* 70.0%

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

   2. +-commutative 70.0%

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

   3. fma-define 70.0%

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

3. Simplified 70.0%

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

5. Taylor expanded in x around 0 67.3%

   \[\leadsto \color{blue}{-1} \]
6. 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 2024103 
(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))))