Kahan p9 Example

Percentage Accurate: 67.6% → 100.0%

Time: 8.5s

Alternatives: 11

Speedup: 2.9×

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: 67.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. add-sqr-sqrt 65.2%

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

   2. times-frac 66.1%

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

   3. hypot-def 66.1%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. log1p-expm1-u 99.9%

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

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right)}\right)\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]
6. Step-by-step derivation

   1. log1p-expm1-u 99.9%

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

   2. associate-*r/ 99.9%

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

   3. associate-/r/ 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. add-sqr-sqrt 65.2%

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

   2. times-frac 66.1%

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

   3. hypot-def 66.1%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 99.9%

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

   3. frac-times 99.9%

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

   4. *-un-lft-identity 99.9%

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

5. Applied egg-rr 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. frac-times 99.9%

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

   3. clear-num 99.9%

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

   4. div-inv 99.7%

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

   5. associate-*l* 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-un-lft-identity 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 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 65.2%

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

   1. add-sqr-sqrt 65.2%

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

   2. times-frac 66.1%

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

   3. hypot-def 66.1%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. add-sqr-sqrt 65.2%

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

   2. times-frac 66.1%

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

   3. hypot-def 66.1%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 99.9%

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

   3. frac-times 99.9%

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

   4. *-un-lft-identity 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 5: 92.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    t_1 = (x / y) * (x / y)
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = t_1 + (t_1 + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double t_1 = (x / y) * (x / y);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (t_1 + -1.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	t_1 = (x / y) * (x / y);
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = t_1 + (t_1 + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(t$95$1 + N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 3.1%

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

      2. fma-def 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. Taylor expanded in y around inf 55.1%

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

      1. associate--l+ 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. times-frac 55.1%

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

      5. distribute-rgt1-in 55.1%

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

      6. metadata-eval 55.1%

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

      7. mul0-lft 55.1%

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

      8. +-commutative 55.1%

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

      9. associate--r+ 55.1%

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

      10. metadata-eval 55.1%

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

      11. unpow2 55.1%

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

      12. unpow2 55.1%

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

      13. associate-*r/ 55.1%

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

      14. associate-*r* 55.1%

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

      15. *-commutative 55.1%

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

      16. times-frac 77.2%

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

   6. Simplified 77.2%

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

4. Final simplification 92.1%

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

Alternative 6: 92.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 3.1%

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

      2. fma-def 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. Taylor expanded in x around 0 56.1%

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

      1. unpow2 56.1%

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

      2. associate-/r* 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in x around 0 55.1%

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

      1. sub-neg 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. metadata-eval 55.1%

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

   9. Simplified 55.1%

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

       1. associate-/l* 55.9%

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

       2. div-inv 55.9%

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

       3. associate-/l* 75.9%

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

   11. Applied egg-rr 75.9%

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

       1. associate-*r/ 76.1%

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

       2. *-rgt-identity 76.1%

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

       3. remove-double-neg 76.1%

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

       4. associate-/r/ 76.1%

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

       5. distribute-rgt-neg-out 76.1%

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

       6. *-commutative 76.1%

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

       7. distribute-lft-neg-out 76.1%

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

       8. remove-double-neg 76.1%

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

   13. Simplified 76.1%

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

4. Final simplification 91.7%

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

Alternative 7: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-153} \lor \neg \left(y \leq 5.5 \cdot 10^{-199}\right):\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.25e-153) (not (<= y 5.5e-199)))
   (+ -1.0 (/ x (* y (/ y x))))
   (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.25e-153) || !(y <= 5.5e-199)) {
		tmp = -1.0 + (x / (y * (y / x)));
	} else {
		tmp = 1.0 + (-2.0 * ((y / 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 <= (-2.25d-153)) .or. (.not. (y <= 5.5d-199))) then
        tmp = (-1.0d0) + (x / (y * (y / x)))
    else
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.25e-153) || !(y <= 5.5e-199)) {
		tmp = -1.0 + (x / (y * (y / x)));
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.25e-153) or not (y <= 5.5e-199):
		tmp = -1.0 + (x / (y * (y / x)))
	else:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.25e-153) || !(y <= 5.5e-199))
		tmp = Float64(-1.0 + Float64(x / Float64(y * Float64(y / x))));
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.25e-153) || ~((y <= 5.5e-199)))
		tmp = -1.0 + (x / (y * (y / x)));
	else
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.25e-153], N[Not[LessEqual[y, 5.5e-199]], $MachinePrecision]], N[(-1.0 + N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.25e-153 or 5.5000000000000001e-199 < y

   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-*r/ 69.3%

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

      2. fma-def 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 84.1%

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

      1. unpow2 84.1%

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

      2. associate-/r* 87.5%

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

   6. Simplified 87.5%

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

   7. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. unpow2 84.2%

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

      3. unpow2 84.2%

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

      4. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

       1. associate-/l* 84.2%

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

       2. div-inv 84.2%

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

       3. associate-/l* 87.7%

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

   11. Applied egg-rr 87.7%

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

       1. associate-*r/ 87.7%

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

       2. *-rgt-identity 87.7%

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

       3. remove-double-neg 87.7%

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

       4. associate-/r/ 87.7%

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

       5. distribute-rgt-neg-out 87.7%

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

       6. *-commutative 87.7%

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

       7. distribute-lft-neg-out 87.7%

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

       8. remove-double-neg 87.7%

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

   13. Simplified 87.7%

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

   if -2.25e-153 < y < 5.5000000000000001e-199

   1. Initial program 56.8%

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

      1. associate-*r/ 57.3%

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

      2. fma-def 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in y around 0 56.8%

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

      1. unpow2 56.8%

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

      2. unpow2 56.8%

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

   6. Simplified 56.8%

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

      1. times-frac 83.9%

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

   8. Applied egg-rr 83.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-153} \lor \neg \left(y \leq 5.5 \cdot 10^{-199}\right):\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 8: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-157} \lor \neg \left(y \leq 2.8 \cdot 10^{-199}\right):\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.2e-157) (not (<= y 2.8e-199)))
   (+ -1.0 (/ x (* y (/ y x))))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.2e-157) || !(y <= 2.8e-199)) {
		tmp = -1.0 + (x / (y * (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 <= (-8.2d-157)) .or. (.not. (y <= 2.8d-199))) then
        tmp = (-1.0d0) + (x / (y * (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 <= -8.2e-157) || !(y <= 2.8e-199)) {
		tmp = -1.0 + (x / (y * (y / x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.2e-157) or not (y <= 2.8e-199):
		tmp = -1.0 + (x / (y * (y / x)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.2e-157) || !(y <= 2.8e-199))
		tmp = Float64(-1.0 + Float64(x / Float64(y * Float64(y / x))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.2e-157) || ~((y <= 2.8e-199)))
		tmp = -1.0 + (x / (y * (y / x)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.2e-157], N[Not[LessEqual[y, 2.8e-199]], $MachinePrecision]], N[(-1.0 + N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -8.2000000000000004e-157 or 2.80000000000000018e-199 < y

   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-*r/ 69.3%

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

      2. fma-def 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 84.1%

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

      1. unpow2 84.1%

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

      2. associate-/r* 87.5%

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

   6. Simplified 87.5%

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

   7. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. unpow2 84.2%

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

      3. unpow2 84.2%

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

      4. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

       1. associate-/l* 84.2%

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

       2. div-inv 84.2%

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

       3. associate-/l* 87.7%

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

   11. Applied egg-rr 87.7%

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

       1. associate-*r/ 87.7%

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

       2. *-rgt-identity 87.7%

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

       3. remove-double-neg 87.7%

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

       4. associate-/r/ 87.7%

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

       5. distribute-rgt-neg-out 87.7%

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

       6. *-commutative 87.7%

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

       7. distribute-lft-neg-out 87.7%

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

       8. remove-double-neg 87.7%

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

   13. Simplified 87.7%

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

   if -8.2000000000000004e-157 < y < 2.80000000000000018e-199

   1. Initial program 56.8%

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

      1. associate-*r/ 57.3%

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

      2. fma-def 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in x around inf 82.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-157} \lor \neg \left(y \leq 2.8 \cdot 10^{-199}\right):\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 82.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-157} \lor \neg \left(y \leq 5.5 \cdot 10^{-199}\right):\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.8e-157) (not (<= y 5.5e-199)))
   (+ -1.0 (/ x (* y (/ y x))))
   (- (+ (/ y x) 1.0) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e-157) || !(y <= 5.5e-199)) {
		tmp = -1.0 + (x / (y * (y / x)));
	} else {
		tmp = ((y / x) + 1.0) - (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 <= (-6.8d-157)) .or. (.not. (y <= 5.5d-199))) then
        tmp = (-1.0d0) + (x / (y * (y / x)))
    else
        tmp = ((y / x) + 1.0d0) - (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e-157) || !(y <= 5.5e-199)) {
		tmp = -1.0 + (x / (y * (y / x)));
	} else {
		tmp = ((y / x) + 1.0) - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.8e-157) or not (y <= 5.5e-199):
		tmp = -1.0 + (x / (y * (y / x)))
	else:
		tmp = ((y / x) + 1.0) - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.8e-157) || !(y <= 5.5e-199))
		tmp = Float64(-1.0 + Float64(x / Float64(y * Float64(y / x))));
	else
		tmp = Float64(Float64(Float64(y / x) + 1.0) - Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.8e-157) || ~((y <= 5.5e-199)))
		tmp = -1.0 + (x / (y * (y / x)));
	else
		tmp = ((y / x) + 1.0) - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.8e-157], N[Not[LessEqual[y, 5.5e-199]], $MachinePrecision]], N[(-1.0 + N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999955e-157 or 5.5000000000000001e-199 < y

   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-*r/ 69.3%

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

      2. fma-def 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 84.1%

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

      1. unpow2 84.1%

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

      2. associate-/r* 87.5%

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

   6. Simplified 87.5%

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

   7. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. unpow2 84.2%

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

      3. unpow2 84.2%

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

      4. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

       1. associate-/l* 84.2%

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

       2. div-inv 84.2%

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

       3. associate-/l* 87.7%

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

   11. Applied egg-rr 87.7%

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

       1. associate-*r/ 87.7%

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

       2. *-rgt-identity 87.7%

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

       3. remove-double-neg 87.7%

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

       4. associate-/r/ 87.7%

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

       5. distribute-rgt-neg-out 87.7%

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

       6. *-commutative 87.7%

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

       7. distribute-lft-neg-out 87.7%

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

       8. remove-double-neg 87.7%

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

   13. Simplified 87.7%

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

   if -6.79999999999999955e-157 < y < 5.5000000000000001e-199

   1. Initial program 56.8%

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

      1. add-sqr-sqrt 56.8%

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

      2. times-frac 57.6%

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

      3. hypot-def 57.6%

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

      4. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 82.5%

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

      1. associate-+r+ 82.5%

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

      2. mul-1-neg 82.5%

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

      3. unsub-neg 82.5%

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

      4. +-commutative 82.5%

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

   6. Simplified 82.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-157} \lor \neg \left(y \leq 5.5 \cdot 10^{-199}\right):\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \end{array} \]

Alternative 10: 82.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.9 \cdot 10^{-155}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-199}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.9e-155) -1.0 (if (<= y 5.5e-199) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.9e-155) {
		tmp = -1.0;
	} else if (y <= 5.5e-199) {
		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 <= (-6.9d-155)) then
        tmp = -1.0d0
    else if (y <= 5.5d-199) 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 <= -6.9e-155) {
		tmp = -1.0;
	} else if (y <= 5.5e-199) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.9e-155:
		tmp = -1.0
	elif y <= 5.5e-199:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.9e-155)
		tmp = -1.0;
	elseif (y <= 5.5e-199)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.9e-155)
		tmp = -1.0;
	elseif (y <= 5.5e-199)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.9e-155], -1.0, If[LessEqual[y, 5.5e-199], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.89999999999999975e-155 or 5.5000000000000001e-199 < y

   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-*r/ 69.3%

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

      2. fma-def 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 87.0%

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

   if -6.89999999999999975e-155 < y < 5.5000000000000001e-199

   1. Initial program 56.8%

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

      1. associate-*r/ 57.3%

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

      2. fma-def 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in x around inf 82.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.9 \cdot 10^{-155}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-199}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11: 66.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 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-*r/ 65.9%

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

   2. fma-def 65.9%

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

3. Simplified 65.9%

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

4. Taylor expanded in x around 0 66.7%

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

5. Final simplification 66.7%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023195 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

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

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