Kahan p9 Example

Percentage Accurate: 67.8% → 100.0%

Time: 11.3s

Alternatives: 13

Speedup: 2.1×

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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (/ (- x y) (hypot x y)) (/ (hypot x y) (+ x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.9%

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

   1. associate-*r/ 65.9%

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

   2. +-commutative 65.9%

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

   3. fma-def 65.9%

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

3. Simplified 65.9%

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

   1. fma-udef 65.9%

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

   2. +-commutative 65.9%

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

   3. *-un-lft-identity 65.9%

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

   4. add-sqr-sqrt 65.9%

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

   5. times-frac 66.0%

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

   6. hypot-def 66.1%

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

   7. hypot-def 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. +-commutative 99.6%

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

7. Simplified 99.6%

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

   1. associate-/l/ 65.9%

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

   2. +-commutative 65.9%

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

   3. clear-num 66.0%

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

   4. *-un-lft-identity 66.0%

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

   5. frac-times 99.7%

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

   6. div-inv 99.9%

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

   7. /-rgt-identity 99.9%

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

   8. associate-/r* 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (* (- x y) (/ (/ (+ x y) (hypot x y)) (hypot x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
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 65.9%

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

   1. associate-*r/ 65.9%

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

   2. +-commutative 65.9%

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

   3. fma-def 65.9%

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

3. Simplified 65.9%

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

   1. fma-udef 65.9%

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

   2. +-commutative 65.9%

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

   3. *-un-lft-identity 65.9%

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

   4. add-sqr-sqrt 65.9%

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

   5. times-frac 66.0%

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

   6. hypot-def 66.1%

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

   7. hypot-def 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. +-commutative 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 92.5% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(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

y = abs(y);
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

NOTE: y should be positive before calling this function
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

y = Math.abs(y);
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

y = abs(y)
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

y = abs(y)
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

y = abs(y)
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

NOTE: y should be positive before calling this function
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 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

      1. fma-udef 3.1%

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

      2. +-commutative 3.1%

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

      3. *-un-lft-identity 3.1%

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

      4. add-sqr-sqrt 3.1%

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

      5. times-frac 3.1%

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

      6. hypot-def 3.1%

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

      7. hypot-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. associate-/l/ 3.1%

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

      2. +-commutative 3.1%

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

      3. clear-num 3.1%

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

      4. *-un-lft-identity 3.1%

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

      5. frac-times 99.7%

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

      6. div-inv 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-/r* 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. clear-num 99.9%

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

       2. inv-pow 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. unpow-1 99.9%

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

   13. Simplified 99.9%

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

   14. Taylor expanded in y around inf 58.6%

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

       1. distribute-rgt1-in 58.6%

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

       2. metadata-eval 58.6%

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

       3. mul0-lft 58.6%

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

       4. associate-+r- 58.6%

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

       5. unpow2 58.6%

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

       6. unpow2 58.6%

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

       7. times-frac 58.6%

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

       8. neg-sub0 58.6%

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

       9. +-commutative 58.6%

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

       10. distribute-neg-in 58.6%

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

       11. metadata-eval 58.6%

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

       12. mul-1-neg 58.6%

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

       13. remove-double-neg 58.6%

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

       14. unpow2 58.6%

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

       15. unpow2 58.6%

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

       16. times-frac 80.6%

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

   16. Simplified 80.6%

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

4. Final simplification 93.3%

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

Alternative 4: 92.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(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) (/ x y)) -1.0))))

C

y = abs(y);
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) * (x / y)) + -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
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) * (x / y)) + -1.0;
	}
	return tmp;
}

Python

y = abs(y)
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) * (x / y)) + -1.0
	return tmp

Julia

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

MATLAB

y = abs(y)
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) * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
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[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around inf 3.1%

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

      1. unpow2 3.1%

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

   6. Simplified 3.1%

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

      1. *-un-lft-identity 3.1%

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

      2. times-frac 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. associate-*l/ 79.8%

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

      2. *-lft-identity 79.8%

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

   10. Simplified 79.8%

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

   11. Taylor expanded in x around 0 58.6%

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

       1. sub-neg 58.6%

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

       2. unpow2 58.6%

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

       3. unpow2 58.6%

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

       4. times-frac 80.3%

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

       5. metadata-eval 80.3%

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

   13. Simplified 80.3%

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

4. Final simplification 93.2%

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

Alternative 5: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-163)
   (- (+ (/ y x) 1.0) (/ y x))
   (if (<= y 1.35e-120)
     (/ (- x y) y)
     (if (<= y 1.25e-111)
       (+ 1.0 (* -2.0 (/ (* y y) (* x x))))
       (+ -1.0 (/ (* x x) (* y y)))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-163) {
		tmp = ((y / x) + 1.0) - (y / x);
	} else if (y <= 1.35e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.25e-111) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.3d-163) then
        tmp = ((y / x) + 1.0d0) - (y / x)
    else if (y <= 1.35d-120) then
        tmp = (x - y) / y
    else if (y <= 1.25d-111) then
        tmp = 1.0d0 + ((-2.0d0) * ((y * y) / (x * x)))
    else
        tmp = (-1.0d0) + ((x * x) / (y * y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-163) {
		tmp = ((y / x) + 1.0) - (y / x);
	} else if (y <= 1.35e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.25e-111) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2.3e-163:
		tmp = ((y / x) + 1.0) - (y / x)
	elif y <= 1.35e-120:
		tmp = (x - y) / y
	elif y <= 1.25e-111:
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)))
	else:
		tmp = -1.0 + ((x * x) / (y * y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-163)
		tmp = Float64(Float64(Float64(y / x) + 1.0) - Float64(y / x));
	elseif (y <= 1.35e-120)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 1.25e-111)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y * y) / Float64(x * x))));
	else
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e-163)
		tmp = ((y / x) + 1.0) - (y / x);
	elseif (y <= 1.35e-120)
		tmp = (x - y) / y;
	elseif (y <= 1.25e-111)
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	else
		tmp = -1.0 + ((x * x) / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 2.3e-163], N[(N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e-120], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.25e-111], N[(1.0 + N[(-2.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.2999999999999999e-163

   1. Initial program 59.0%

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

      1. associate-*r/ 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-def 59.2%

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

   3. Simplified 59.2%

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

      1. fma-udef 59.2%

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

      2. +-commutative 59.2%

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

      3. *-un-lft-identity 59.2%

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

      4. add-sqr-sqrt 59.2%

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

      5. times-frac 59.3%

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

      6. hypot-def 59.3%

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

      7. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)} \]
   6. 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. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-/l/ 59.2%

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

      2. +-commutative 59.2%

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

      3. clear-num 59.2%

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

      4. *-un-lft-identity 59.2%

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

      5. frac-times 99.7%

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

      6. div-inv 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-/r* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around inf 30.3%

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

       1. associate-+r+ 30.3%

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

       2. +-commutative 30.3%

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

       3. associate-*r/ 30.3%

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

       4. neg-mul-1 30.3%

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

   12. Simplified 30.3%

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

       1. add-sqr-sqrt 29.1%

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

       2. distribute-frac-neg 29.1%

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

       3. fma-def 29.1%

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

       4. add-sqr-sqrt 13.0%

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

       5. sqrt-unprod 28.5%

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

       6. sqr-neg 28.5%

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

       7. sqrt-unprod 15.5%

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

       8. add-sqr-sqrt 28.1%

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

       9. fma-neg 28.1%

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

       10. add-sqr-sqrt 31.1%

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

       11. add-sqr-sqrt 18.5%

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

       12. sqrt-unprod 31.4%

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

       13. sqr-neg 31.4%

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

       14. sqrt-unprod 13.0%

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

       15. add-sqr-sqrt 30.3%

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

   14. Applied egg-rr 30.3%

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

   if 2.2999999999999999e-163 < y < 1.3499999999999999e-120

   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* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 56.8%

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

   if 1.3499999999999999e-120 < y < 1.2500000000000001e-111

   1. Initial program 100.0%

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

      1. associate-*r/ 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 1.2500000000000001e-111 < y

   1. Initial program 99.8%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. unpow2 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in x around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. unpow2 85.9%

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

      3. unpow2 85.9%

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

      4. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 6: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-163}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\frac{1}{x} + \frac{\frac{y}{x}}{x}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-163)
   (* (- x y) (+ (/ 1.0 x) (/ (/ y x) x)))
   (if (<= y 1.4e-120)
     (/ (- x y) y)
     (if (<= y 1.15e-111)
       (+ 1.0 (* -2.0 (/ (* y y) (* x x))))
       (+ -1.0 (/ (* x x) (* y y)))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-163) {
		tmp = (x - y) * ((1.0 / x) + ((y / x) / x));
	} else if (y <= 1.4e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.15e-111) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.5d-163) then
        tmp = (x - y) * ((1.0d0 / x) + ((y / x) / x))
    else if (y <= 1.4d-120) then
        tmp = (x - y) / y
    else if (y <= 1.15d-111) then
        tmp = 1.0d0 + ((-2.0d0) * ((y * y) / (x * x)))
    else
        tmp = (-1.0d0) + ((x * x) / (y * y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-163) {
		tmp = (x - y) * ((1.0 / x) + ((y / x) / x));
	} else if (y <= 1.4e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.15e-111) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5.5e-163:
		tmp = (x - y) * ((1.0 / x) + ((y / x) / x))
	elif y <= 1.4e-120:
		tmp = (x - y) / y
	elif y <= 1.15e-111:
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)))
	else:
		tmp = -1.0 + ((x * x) / (y * y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-163)
		tmp = Float64(Float64(x - y) * Float64(Float64(1.0 / x) + Float64(Float64(y / x) / x)));
	elseif (y <= 1.4e-120)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 1.15e-111)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y * y) / Float64(x * x))));
	else
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-163)
		tmp = (x - y) * ((1.0 / x) + ((y / x) / x));
	elseif (y <= 1.4e-120)
		tmp = (x - y) / y;
	elseif (y <= 1.15e-111)
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	else
		tmp = -1.0 + ((x * x) / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5.5e-163], N[(N[(x - y), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-120], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.15e-111], N[(1.0 + N[(-2.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.4999999999999998e-163

   1. Initial program 59.0%

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

      1. associate-*r/ 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-def 59.2%

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

   3. Simplified 59.2%

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

      1. fma-udef 59.2%

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

      2. +-commutative 59.2%

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

      3. *-un-lft-identity 59.2%

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

      4. add-sqr-sqrt 59.2%

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

      5. times-frac 59.3%

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

      6. hypot-def 59.3%

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

      7. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)} \]
   6. 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. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 32.3%

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

   9. Taylor expanded in y around 0 23.9%

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

       1. +-commutative 23.9%

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

       2. unpow2 23.9%

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

       3. associate-/r* 31.7%

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

   11. Simplified 31.7%

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

   if 5.4999999999999998e-163 < y < 1.39999999999999997e-120

   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* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 56.8%

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

   if 1.39999999999999997e-120 < y < 1.15e-111

   1. Initial program 100.0%

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

      1. associate-*r/ 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 1.15e-111 < y

   1. Initial program 99.8%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. unpow2 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in x around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. unpow2 85.9%

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

      3. unpow2 85.9%

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

      4. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-163}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\frac{1}{x} + \frac{\frac{y}{x}}{x}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 7: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-167}:\\ \;\;\;\;\frac{x - y}{\left(x - y\right) + y \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 4.9e-167)
   (/ (- x y) (+ (- x y) (* y (/ y x))))
   (if (<= y 1.4e-120)
     (/ (- x y) y)
     (if (<= y 1.15e-111)
       (+ 1.0 (* -2.0 (/ (* y y) (* x x))))
       (+ -1.0 (/ (* x x) (* y y)))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-167) {
		tmp = (x - y) / ((x - y) + (y * (y / x)));
	} else if (y <= 1.4e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.15e-111) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.9d-167) then
        tmp = (x - y) / ((x - y) + (y * (y / x)))
    else if (y <= 1.4d-120) then
        tmp = (x - y) / y
    else if (y <= 1.15d-111) then
        tmp = 1.0d0 + ((-2.0d0) * ((y * y) / (x * x)))
    else
        tmp = (-1.0d0) + ((x * x) / (y * y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-167) {
		tmp = (x - y) / ((x - y) + (y * (y / x)));
	} else if (y <= 1.4e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.15e-111) {
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 4.9e-167:
		tmp = (x - y) / ((x - y) + (y * (y / x)))
	elif y <= 1.4e-120:
		tmp = (x - y) / y
	elif y <= 1.15e-111:
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)))
	else:
		tmp = -1.0 + ((x * x) / (y * y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 4.9e-167)
		tmp = Float64(Float64(x - y) / Float64(Float64(x - y) + Float64(y * Float64(y / x))));
	elseif (y <= 1.4e-120)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 1.15e-111)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y * y) / Float64(x * x))));
	else
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.9e-167)
		tmp = (x - y) / ((x - y) + (y * (y / x)));
	elseif (y <= 1.4e-120)
		tmp = (x - y) / y;
	elseif (y <= 1.15e-111)
		tmp = 1.0 + (-2.0 * ((y * y) / (x * x)));
	else
		tmp = -1.0 + ((x * x) / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 4.9e-167], N[(N[(x - y), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] + N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-120], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.15e-111], N[(1.0 + N[(-2.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.90000000000000003e-167

   1. Initial program 59.0%

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

      1. associate-/l* 59.4%

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

      2. fma-def 59.4%

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

   3. Simplified 59.4%

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

      1. fma-def 59.4%

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

      2. add-sqr-sqrt 59.4%

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

      3. *-un-lft-identity 59.4%

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

      4. times-frac 59.4%

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

      5. hypot-def 59.5%

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

      6. hypot-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 30.8%

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

      1. cancel-sign-sub-inv 30.8%

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

      2. +-commutative 30.8%

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

      3. neg-mul-1 30.8%

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

      4. +-commutative 30.8%

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

      5. sub-neg 30.8%

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

      6. unpow2 30.8%

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

      7. associate-*r/ 31.2%

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

      8. metadata-eval 31.2%

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

      9. unpow2 31.2%

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

      10. associate-*r/ 31.4%

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

      11. *-lft-identity 31.4%

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

   8. Simplified 31.4%

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

   9. Taylor expanded in x around inf 31.1%

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

       1. +-commutative 31.1%

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

       2. mul-1-neg 31.1%

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

       3. sub-neg 31.1%

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

   11. Simplified 31.1%

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

   if 4.90000000000000003e-167 < y < 1.39999999999999997e-120

   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* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 56.8%

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

   if 1.39999999999999997e-120 < y < 1.15e-111

   1. Initial program 100.0%

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

      1. associate-*r/ 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 1.15e-111 < y

   1. Initial program 99.8%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. unpow2 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in x around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. unpow2 85.9%

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

      3. unpow2 85.9%

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

      4. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-167}:\\ \;\;\;\;\frac{x - y}{\left(x - y\right) + y \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 8: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e-168)
   1.0
   (if (<= y 5.8e-121)
     (/ (- x y) y)
     (if (<= y 1.1e-111) 1.0 (+ (* (/ x y) (/ x y)) -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-168) {
		tmp = 1.0;
	} else if (y <= 5.8e-121) {
		tmp = (x - y) / y;
	} else if (y <= 1.1e-111) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) * (x / y)) + -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.5d-168) then
        tmp = 1.0d0
    else if (y <= 5.8d-121) then
        tmp = (x - y) / y
    else if (y <= 1.1d-111) then
        tmp = 1.0d0
    else
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-168) {
		tmp = 1.0;
	} else if (y <= 5.8e-121) {
		tmp = (x - y) / y;
	} else if (y <= 1.1e-111) {
		tmp = 1.0;
	} else {
		tmp = ((x / y) * (x / y)) + -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 7.5e-168:
		tmp = 1.0
	elif y <= 5.8e-121:
		tmp = (x - y) / y
	elif y <= 1.1e-111:
		tmp = 1.0
	else:
		tmp = ((x / y) * (x / y)) + -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 7.5e-168)
		tmp = 1.0;
	elseif (y <= 5.8e-121)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 1.1e-111)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.5e-168)
		tmp = 1.0;
	elseif (y <= 5.8e-121)
		tmp = (x - y) / y;
	elseif (y <= 1.1e-111)
		tmp = 1.0;
	else
		tmp = ((x / y) * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 7.5e-168], 1.0, If[LessEqual[y, 5.8e-121], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.1e-111], 1.0, N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.4999999999999995e-168 or 5.8e-121 < y < 1.1e-111

   1. Initial program 59.2%

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

      1. associate-*r/ 59.3%

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

      2. +-commutative 59.3%

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

      3. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 30.4%

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

   if 7.4999999999999995e-168 < y < 5.8e-121

   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* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 56.8%

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

   if 1.1e-111 < y

   1. Initial program 99.8%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. unpow2 85.4%

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

   6. Simplified 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. times-frac 85.5%

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

   8. Applied egg-rr 85.5%

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

      1. associate-*l/ 85.5%

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

      2. *-lft-identity 85.5%

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

   10. Simplified 85.5%

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

   11. Taylor expanded in x around 0 85.9%

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

       1. sub-neg 85.9%

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

       2. unpow2 85.9%

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

       3. unpow2 85.9%

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

       4. times-frac 85.9%

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

       5. metadata-eval 85.9%

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

   13. Simplified 85.9%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 9: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-162)
   1.0
   (if (<= y 1.38e-120)
     (/ (- x y) y)
     (if (<= y 1.1e-111) 1.0 (+ -1.0 (/ (* x x) (* y y)))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-162) {
		tmp = 1.0;
	} else if (y <= 1.38e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.1e-111) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-162) then
        tmp = 1.0d0
    else if (y <= 1.38d-120) then
        tmp = (x - y) / y
    else if (y <= 1.1d-111) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + ((x * x) / (y * y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-162) {
		tmp = 1.0;
	} else if (y <= 1.38e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.1e-111) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.3e-162:
		tmp = 1.0
	elif y <= 1.38e-120:
		tmp = (x - y) / y
	elif y <= 1.1e-111:
		tmp = 1.0
	else:
		tmp = -1.0 + ((x * x) / (y * y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-162)
		tmp = 1.0;
	elseif (y <= 1.38e-120)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 1.1e-111)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-162)
		tmp = 1.0;
	elseif (y <= 1.38e-120)
		tmp = (x - y) / y;
	elseif (y <= 1.1e-111)
		tmp = 1.0;
	else
		tmp = -1.0 + ((x * x) / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.3e-162], 1.0, If[LessEqual[y, 1.38e-120], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.1e-111], 1.0, N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3e-162 or 1.38e-120 < y < 1.1e-111

   1. Initial program 59.2%

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

      1. associate-*r/ 59.3%

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

      2. +-commutative 59.3%

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

      3. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 30.4%

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

   if 1.3e-162 < y < 1.38e-120

   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* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 56.8%

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

   if 1.1e-111 < y

   1. Initial program 99.8%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. unpow2 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in x around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. unpow2 85.9%

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

      3. unpow2 85.9%

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

      4. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 10: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-169)
   (- (+ (/ y x) 1.0) (/ y x))
   (if (<= y 1.4e-120)
     (/ (- x y) y)
     (if (<= y 1.1e-111) 1.0 (+ -1.0 (/ (* x x) (* y y)))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-169) {
		tmp = ((y / x) + 1.0) - (y / x);
	} else if (y <= 1.4e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.1e-111) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d-169) then
        tmp = ((y / x) + 1.0d0) - (y / x)
    else if (y <= 1.4d-120) then
        tmp = (x - y) / y
    else if (y <= 1.1d-111) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + ((x * x) / (y * y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-169) {
		tmp = ((y / x) + 1.0) - (y / x);
	} else if (y <= 1.4e-120) {
		tmp = (x - y) / y;
	} else if (y <= 1.1e-111) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2.4e-169:
		tmp = ((y / x) + 1.0) - (y / x)
	elif y <= 1.4e-120:
		tmp = (x - y) / y
	elif y <= 1.1e-111:
		tmp = 1.0
	else:
		tmp = -1.0 + ((x * x) / (y * y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-169)
		tmp = Float64(Float64(Float64(y / x) + 1.0) - Float64(y / x));
	elseif (y <= 1.4e-120)
		tmp = Float64(Float64(x - y) / y);
	elseif (y <= 1.1e-111)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4e-169)
		tmp = ((y / x) + 1.0) - (y / x);
	elseif (y <= 1.4e-120)
		tmp = (x - y) / y;
	elseif (y <= 1.1e-111)
		tmp = 1.0;
	else
		tmp = -1.0 + ((x * x) / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 2.4e-169], N[(N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-120], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.1e-111], 1.0, N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.40000000000000011e-169

   1. Initial program 59.0%

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

      1. associate-*r/ 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-def 59.2%

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

   3. Simplified 59.2%

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

      1. fma-udef 59.2%

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

      2. +-commutative 59.2%

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

      3. *-un-lft-identity 59.2%

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

      4. add-sqr-sqrt 59.2%

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

      5. times-frac 59.3%

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

      6. hypot-def 59.3%

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

      7. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)} \]
   6. 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. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-/l/ 59.2%

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

      2. +-commutative 59.2%

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

      3. clear-num 59.2%

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

      4. *-un-lft-identity 59.2%

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

      5. frac-times 99.7%

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

      6. div-inv 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-/r* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around inf 30.3%

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

       1. associate-+r+ 30.3%

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

       2. +-commutative 30.3%

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

       3. associate-*r/ 30.3%

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

       4. neg-mul-1 30.3%

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

   12. Simplified 30.3%

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

       1. add-sqr-sqrt 29.1%

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

       2. distribute-frac-neg 29.1%

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

       3. fma-def 29.1%

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

       4. add-sqr-sqrt 13.0%

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

       5. sqrt-unprod 28.5%

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

       6. sqr-neg 28.5%

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

       7. sqrt-unprod 15.5%

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

       8. add-sqr-sqrt 28.1%

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

       9. fma-neg 28.1%

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

       10. add-sqr-sqrt 31.1%

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

       11. add-sqr-sqrt 18.5%

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

       12. sqrt-unprod 31.4%

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

       13. sqr-neg 31.4%

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

       14. sqrt-unprod 13.0%

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

       15. add-sqr-sqrt 30.3%

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

   14. Applied egg-rr 30.3%

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

   if 2.40000000000000011e-169 < y < 1.39999999999999997e-120

   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* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 56.8%

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

   if 1.39999999999999997e-120 < y < 1.1e-111

   1. Initial program 100.0%

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

      1. associate-*r/ 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around inf 100.0%

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

   if 1.1e-111 < y

   1. Initial program 99.8%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. unpow2 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in x around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. unpow2 85.9%

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

      3. unpow2 85.9%

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

      4. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 11: 83.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-164)
   1.0
   (if (<= y 1.4e-120) -1.0 (if (<= y 1.7e-111) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-164) {
		tmp = 1.0;
	} else if (y <= 1.4e-120) {
		tmp = -1.0;
	} else if (y <= 1.7e-111) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.8d-164) then
        tmp = 1.0d0
    else if (y <= 1.4d-120) then
        tmp = -1.0d0
    else if (y <= 1.7d-111) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-164) {
		tmp = 1.0;
	} else if (y <= 1.4e-120) {
		tmp = -1.0;
	} else if (y <= 1.7e-111) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 4.8e-164:
		tmp = 1.0
	elif y <= 1.4e-120:
		tmp = -1.0
	elif y <= 1.7e-111:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-164)
		tmp = 1.0;
	elseif (y <= 1.4e-120)
		tmp = -1.0;
	elseif (y <= 1.7e-111)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-164)
		tmp = 1.0;
	elseif (y <= 1.4e-120)
		tmp = -1.0;
	elseif (y <= 1.7e-111)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 4.8e-164], 1.0, If[LessEqual[y, 1.4e-120], -1.0, If[LessEqual[y, 1.7e-111], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.79999999999999966e-164 or 1.39999999999999997e-120 < y < 1.69999999999999998e-111

   1. Initial program 59.2%

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

      1. associate-*r/ 59.3%

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

      2. +-commutative 59.3%

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

      3. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 30.4%

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

   if 4.79999999999999966e-164 < y < 1.39999999999999997e-120 or 1.69999999999999998e-111 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 75.7%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12: 83.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 1.45e-162) 1.0 (/ (- x y) y)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-162) {
		tmp = 1.0;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.45d-162) then
        tmp = 1.0d0
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-162) {
		tmp = 1.0;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.45e-162:
		tmp = 1.0
	else:
		tmp = (x - y) / y
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-162)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.45e-162)
		tmp = 1.0;
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.45e-162], 1.0, N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4500000000000001e-162

   1. Initial program 59.0%

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

      1. associate-*r/ 59.2%

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

      2. +-commutative 59.2%

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

      3. fma-def 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around inf 30.1%

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

   if 1.4500000000000001e-162 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 74.0%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 13: 66.9% accurate, 15.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ -1 \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	return -1.0;
}

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

y = abs(y)
function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 65.9%

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

   1. associate-*r/ 65.9%

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

   2. +-commutative 65.9%

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

   3. fma-def 65.9%

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

3. Simplified 65.9%

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

4. Taylor expanded in x around 0 69.9%

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

5. Final simplification 69.9%

   \[\leadsto -1 \]

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 2023240 
(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))))