Kahan p9 Example

Percentage Accurate: 67.6% → 100.0%

Time: 10.7s

Alternatives: 12

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: 67.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

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 66.0%

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

   1. fma-def 66.0%

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

   2. add-sqr-sqrt 66.0%

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

   3. times-frac 66.5%

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

   4. fma-def 66.5%

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

   5. hypot-def 66.5%

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

   6. fma-def 66.5%

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

   7. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

6. 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 66.0%

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

   1. +-commutative 66.0%

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

   2. +-commutative 66.0%

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

   3. associate-*r/ 66.3%

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

   4. +-commutative 66.3%

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

   5. +-commutative 66.3%

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

   6. fma-def 66.3%

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

3. Simplified 66.3%

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

   1. add-sqr-sqrt 66.3%

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

   2. *-un-lft-identity 66.3%

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

   3. times-frac 66.3%

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

   4. fma-def 66.3%

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

   5. hypot-def 66.3%

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

   6. fma-def 66.3%

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

   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. Final simplification 99.7%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

y = abs(y)
function tmp = code(x, y)
	tmp = ((x - y) / hypot(x, y)) * ((x + y) / hypot(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[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

   1. fma-def 66.0%

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

   2. add-sqr-sqrt 66.0%

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

   3. times-frac 66.5%

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

   4. fma-def 66.5%

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

   5. hypot-def 66.5%

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

   6. fma-def 66.5%

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

   7. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. fma-def 66.0%

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

   2. add-sqr-sqrt 66.0%

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

   3. times-frac 66.5%

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

   4. fma-def 66.5%

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

   5. hypot-def 66.5%

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

   6. fma-def 66.5%

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

   7. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. clear-num 99.9%

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

   3. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 5: 93.1% accurate, 0.5× speedup

Math

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

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

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) / ((y + (x / (y / x))) - x);
	}
	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) / ((y + (x / (y / x))) - x)
    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) / ((y + (x / (y / x))) - x);
	}
	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) / ((y + (x / (y / x))) - x)
	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(x - y) / Float64(Float64(y + Float64(x / Float64(y / x))) - x));
	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) / ((y + (x / (y / x))) - x);
	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[(x - y), $MachinePrecision] / N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $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 99.9%

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

   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-/l* 3.1%

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

      2. +-commutative 3.1%

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

      3. remove-double-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

      7. sub-neg 3.1%

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

      8. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 3.1%

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

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

   7. Taylor expanded in y around inf 71.9%

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

      1. +-commutative 71.9%

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

      2. neg-mul-1 71.9%

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

      3. +-commutative 71.9%

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

      4. associate-+l+ 71.9%

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

      5. unpow2 71.9%

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

      6. +-commutative 71.9%

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

      7. sub-neg 71.9%

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

   9. Simplified 71.9%

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

       1. associate-+r- 71.9%

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

       2. associate-/l* 73.3%

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

   11. Applied egg-rr 73.3%

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

4. Final simplification 90.9%

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

Alternative 6: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-162}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{elif}\;y \leq 10000:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-162)
   (+ 1.0 (* -2.0 (/ (/ y (/ x y)) x)))
   (if (<= y 10000.0) (* (- x y) (/ (+ x y) (+ (* x x) (* y y)))) -1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-162) {
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x));
	} else if (y <= 10000.0) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	} 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 <= 1.85d-162) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / (x / y)) / x))
    else if (y <= 10000.0d0) then
        tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
    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 <= 1.85e-162) {
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x));
	} else if (y <= 10000.0) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.85e-162:
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x))
	elif y <= 10000.0:
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-162)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / Float64(x / y)) / x)));
	elseif (y <= 10000.0)
		tmp = Float64(Float64(x - y) * Float64(Float64(x + y) / Float64(Float64(x * x) + Float64(y * y))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-162)
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x));
	elseif (y <= 10000.0)
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	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, 1.85e-162], N[(1.0 + N[(-2.0 * N[(N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 10000.0], N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8500000000000001e-162

   1. Initial program 57.1%

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

      1. +-commutative 57.1%

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

      2. +-commutative 57.1%

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

      3. associate-*r/ 58.3%

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

      4. +-commutative 58.3%

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

      5. +-commutative 58.3%

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

      6. fma-def 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in y around 0 26.6%

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

      1. unpow2 26.6%

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

      2. unpow2 26.6%

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

   6. Simplified 26.6%

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

      1. *-un-lft-identity 26.6%

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

      2. times-frac 37.5%

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

   8. Applied egg-rr 37.5%

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

      1. associate-*l/ 37.5%

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

      2. *-lft-identity 37.5%

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

      3. associate-/l* 38.5%

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

   10. Simplified 38.5%

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

   if 1.8500000000000001e-162 < y < 1e4

   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. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-*r/ 96.8%

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

      4. +-commutative 96.8%

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

      5. +-commutative 96.8%

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

      6. fma-def 96.8%

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

   3. Simplified 96.8%

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

      1. fma-def 96.8%

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

      2. +-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

   if 1e4 < y

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-*r/ 66.3%

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

      4. +-commutative 66.3%

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

      5. +-commutative 66.3%

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

      6. fma-def 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in x around 0 65.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-162}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{elif}\;y \leq 10000:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-132} \lor \neg \left(y \leq 3.8 \cdot 10^{-101}\right):\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-162)
   1.0
   (if (or (<= y 1.1e-132) (not (<= y 3.8e-101)))
     (+ -1.0 (/ (* x x) (* y y)))
     (/ (- x y) x))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-162) {
		tmp = 1.0;
	} else if ((y <= 1.1e-132) || !(y <= 3.8e-101)) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else {
		tmp = (x - y) / x;
	}
	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-162) then
        tmp = 1.0d0
    else if ((y <= 1.1d-132) .or. (.not. (y <= 3.8d-101))) then
        tmp = (-1.0d0) + ((x * x) / (y * y))
    else
        tmp = (x - y) / x
    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-162) {
		tmp = 1.0;
	} else if ((y <= 1.1e-132) || !(y <= 3.8e-101)) {
		tmp = -1.0 + ((x * x) / (y * y));
	} else {
		tmp = (x - y) / x;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2.3e-162:
		tmp = 1.0
	elif (y <= 1.1e-132) or not (y <= 3.8e-101):
		tmp = -1.0 + ((x * x) / (y * y))
	else:
		tmp = (x - y) / x
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-162)
		tmp = 1.0;
	elseif ((y <= 1.1e-132) || !(y <= 3.8e-101))
		tmp = Float64(-1.0 + Float64(Float64(x * x) / Float64(y * y)));
	else
		tmp = Float64(Float64(x - y) / x);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e-162)
		tmp = 1.0;
	elseif ((y <= 1.1e-132) || ~((y <= 3.8e-101)))
		tmp = -1.0 + ((x * x) / (y * y));
	else
		tmp = (x - y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 2.3e-162], 1.0, If[Or[LessEqual[y, 1.1e-132], N[Not[LessEqual[y, 3.8e-101]], $MachinePrecision]], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.2999999999999998e-162

   1. Initial program 57.1%

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

      1. +-commutative 57.1%

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

      2. +-commutative 57.1%

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

      3. associate-*r/ 58.3%

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

      4. +-commutative 58.3%

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

      5. +-commutative 58.3%

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

      6. fma-def 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in x around inf 36.3%

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

   if 2.2999999999999998e-162 < y < 1.09999999999999995e-132 or 3.8000000000000001e-101 < 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. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r/ 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. fma-def 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in x around 0 81.8%

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

      1. unpow2 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in x around 0 85.5%

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

      1. sub-neg 85.5%

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

      2. metadata-eval 85.5%

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

      3. unpow2 85.5%

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

      4. unpow2 85.5%

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

   9. Simplified 85.5%

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

   if 1.09999999999999995e-132 < y < 3.8000000000000001e-101

   1. Initial program 99.6%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. sub-neg 99.7%

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

      8. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 58.9%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-132} \lor \neg \left(y \leq 3.8 \cdot 10^{-101}\right):\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \]

Alternative 8: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x}{y} + \left(y - x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-111)
   (+ 1.0 (* -2.0 (/ (/ y (/ x y)) x)))
   (/ (- x y) (+ (/ (* x x) y) (- y x)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-111) {
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x));
	} else {
		tmp = (x - y) / (((x * x) / y) + (y - x));
	}
	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-111) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / (x / y)) / x))
    else
        tmp = (x - y) / (((x * x) / y) + (y - x))
    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-111) {
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x));
	} else {
		tmp = (x - y) / (((x * x) / y) + (y - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.45e-111:
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x))
	else:
		tmp = (x - y) / (((x * x) / y) + (y - x))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-111)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / Float64(x / y)) / x)));
	else
		tmp = Float64(Float64(x - y) / Float64(Float64(Float64(x * x) / y) + Float64(y - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.45e-111)
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x));
	else
		tmp = (x - y) / (((x * x) / y) + (y - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.45e-111], N[(1.0 + N[(-2.0 * N[(N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] + N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45000000000000001e-111

   1. Initial program 60.0%

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

      1. +-commutative 60.0%

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

      2. +-commutative 60.0%

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

      3. associate-*r/ 60.4%

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

      4. +-commutative 60.4%

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

      5. +-commutative 60.4%

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

      6. fma-def 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in y around 0 27.7%

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

      1. unpow2 27.7%

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

      2. unpow2 27.7%

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

   6. Simplified 27.7%

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

      1. *-un-lft-identity 27.7%

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

      2. times-frac 37.9%

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

   8. Applied egg-rr 37.9%

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

      1. associate-*l/ 37.9%

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

      2. *-lft-identity 37.9%

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

      3. associate-/l* 38.8%

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

   10. Simplified 38.8%

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

   if 1.45000000000000001e-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-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. remove-double-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. sub-neg 99.8%

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

      8. remove-double-neg 99.8%

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

   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 83.8%

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

      1. unpow2 83.7%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in y around inf 84.2%

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

      1. +-commutative 84.2%

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

      2. neg-mul-1 84.2%

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

      3. +-commutative 84.2%

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

      4. associate-+l+ 84.2%

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

      5. unpow2 84.2%

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

      6. +-commutative 84.2%

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

      7. sub-neg 84.2%

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

   9. Simplified 84.2%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x}{y} + \left(y - x\right)}\\ \end{array} \]

Alternative 9: 82.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-111)
   (+ 1.0 (* -2.0 (/ (/ y (/ x y)) x)))
   (+ -1.0 (/ (* x x) (* y y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-111) {
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / 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 <= 1.45d-111) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / (x / y)) / 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 <= 1.45e-111) {
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x));
	} else {
		tmp = -1.0 + ((x * x) / (y * y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.45e-111:
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / x))
	else:
		tmp = -1.0 + ((x * x) / (y * y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-111)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / Float64(x / y)) / 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 <= 1.45e-111)
		tmp = 1.0 + (-2.0 * ((y / (x / y)) / 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, 1.45e-111], N[(1.0 + N[(-2.0 * N[(N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45000000000000001e-111

   1. Initial program 60.0%

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

      1. +-commutative 60.0%

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

      2. +-commutative 60.0%

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

      3. associate-*r/ 60.4%

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

      4. +-commutative 60.4%

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

      5. +-commutative 60.4%

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

      6. fma-def 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in y around 0 27.7%

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

      1. unpow2 27.7%

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

      2. unpow2 27.7%

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

   6. Simplified 27.7%

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

      1. *-un-lft-identity 27.7%

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

      2. times-frac 37.9%

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

   8. Applied egg-rr 37.9%

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

      1. associate-*l/ 37.9%

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

      2. *-lft-identity 37.9%

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

      3. associate-/l* 38.8%

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

   10. Simplified 38.8%

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

   if 1.45000000000000001e-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. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r/ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 83.7%

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

      1. unpow2 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in x around 0 84.0%

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

      1. sub-neg 84.0%

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

      2. metadata-eval 84.0%

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

      3. unpow2 84.0%

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

      4. unpow2 84.0%

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

   9. Simplified 84.0%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-111}:\\ \;\;\;\;1 + -2 \cdot \frac{\frac{y}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 10: 81.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-132}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.95e-162) 1.0 (if (<= y 1e-132) -1.0 (if (<= y 2e-111) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-162) {
		tmp = 1.0;
	} else if (y <= 1e-132) {
		tmp = -1.0;
	} else if (y <= 2e-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 <= 1.95d-162) then
        tmp = 1.0d0
    else if (y <= 1d-132) then
        tmp = -1.0d0
    else if (y <= 2d-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 <= 1.95e-162) {
		tmp = 1.0;
	} else if (y <= 1e-132) {
		tmp = -1.0;
	} else if (y <= 2e-111) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.95e-162:
		tmp = 1.0
	elif y <= 1e-132:
		tmp = -1.0
	elif y <= 2e-111:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.95e-162)
		tmp = 1.0;
	elseif (y <= 1e-132)
		tmp = -1.0;
	elseif (y <= 2e-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 <= 1.95e-162)
		tmp = 1.0;
	elseif (y <= 1e-132)
		tmp = -1.0;
	elseif (y <= 2e-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, 1.95e-162], 1.0, If[LessEqual[y, 1e-132], -1.0, If[LessEqual[y, 2e-111], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.95e-162 or 9.9999999999999999e-133 < y < 2.00000000000000018e-111

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. +-commutative 58.3%

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

      3. associate-*r/ 59.5%

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

      4. +-commutative 59.5%

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

      5. +-commutative 59.5%

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

      6. fma-def 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in x around inf 36.7%

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

   if 1.95e-162 < y < 9.9999999999999999e-133 or 2.00000000000000018e-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. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r/ 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. fma-def 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in x around 0 79.7%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-132}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11: 81.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-101}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 3.7e-101) (/ (- x y) x) -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-101) {
		tmp = (x - y) / x;
	} 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 <= 3.7d-101) then
        tmp = (x - y) / x
    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 <= 3.7e-101) {
		tmp = (x - y) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 3.7e-101:
		tmp = (x - y) / x
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 3.7e-101)
		tmp = Float64(Float64(x - y) / x);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.7e-101)
		tmp = (x - y) / x;
	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, 3.7e-101], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.70000000000000005e-101

   1. Initial program 60.9%

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

      1. associate-/l* 61.5%

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

      2. +-commutative 61.5%

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

      3. remove-double-neg 61.5%

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

      4. sub-neg 61.5%

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

      5. +-commutative 61.5%

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

      6. fma-def 61.5%

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

      7. sub-neg 61.5%

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

      8. remove-double-neg 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in x around inf 36.0%

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

   if 3.70000000000000005e-101 < 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. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r/ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 89.0%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-101}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12: 67.2% 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 66.0%

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

   1. +-commutative 66.0%

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

   2. +-commutative 66.0%

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

   3. associate-*r/ 66.3%

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

   4. +-commutative 66.3%

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

   5. +-commutative 66.3%

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

   6. fma-def 66.3%

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

3. Simplified 66.3%

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

4. Taylor expanded in x around 0 65.0%

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

5. Final simplification 65.0%

   \[\leadsto -1 \]

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 2023270 
(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))))