Kahan p9 Example

Percentage Accurate: 69.1% → 100.0%

Time: 9.6s

Alternatives: 10

Speedup: 0.4×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-*r/ 71.0%

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

   3. +-commutative 71.0%

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

   4. fma-def 71.0%

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

3. Simplified 71.0%

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

   1. *-un-lft-identity 71.0%

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

   2. add-sqr-sqrt 71.0%

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

   3. times-frac 71.1%

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

   4. fma-def 71.1%

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

   5. hypot-def 71.1%

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

   6. fma-def 71.1%

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

   7. hypot-def 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. div-inv 99.9%

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

   3. clear-num 100.0%

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

   4. un-div-inv 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. fma-def 71.1%

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

   2. add-sqr-sqrt 71.1%

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

   3. times-frac 71.3%

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

   4. fma-def 71.3%

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

   5. hypot-def 71.3%

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

   6. fma-def 71.3%

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

   7. hypot-def 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-*r/ 71.0%

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

   3. +-commutative 71.0%

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

   4. fma-def 71.0%

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

3. Simplified 71.0%

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

   1. *-un-lft-identity 71.0%

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

   2. add-sqr-sqrt 71.0%

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

   3. times-frac 71.1%

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

   4. fma-def 71.1%

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

   5. hypot-def 71.1%

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

   6. fma-def 71.1%

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

   7. hypot-def 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. div-inv 99.9%

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

   3. associate-*l/ 100.0%

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

   4. associate-/l* 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 4: 92.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(2.0 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $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 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r/ 3.1%

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

      3. +-commutative 3.1%

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

      4. fma-def 3.1%

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

   3. Simplified 3.1%

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

      1. *-un-lft-identity 3.1%

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

      2. add-sqr-sqrt 3.1%

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

      3. times-frac 3.1%

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

      4. fma-def 3.1%

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

      5. hypot-def 3.1%

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

      6. fma-def 3.1%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r* 99.7%

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

      2. div-inv 99.9%

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

      3. associate-*l/ 99.9%

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

      4. associate-/l* 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 54.1%

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

       1. fma-neg 54.1%

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

       2. unpow2 54.1%

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

       3. unpow2 54.1%

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

       4. times-frac 80.4%

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

       5. unpow2 80.4%

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

       6. metadata-eval 80.4%

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

   11. Simplified 80.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]
   12. Step-by-step derivation

       1. unpow2 80.4%

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

       2. clear-num 80.4%

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

       3. un-div-inv 80.4%

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

   13. Applied egg-rr 80.4%

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

4. Final simplification 94.3%

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

Alternative 5: 92.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing

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

   1. Initial program 0.0%

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

      1. fma-def 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 3.1%

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

      4. fma-def 3.1%

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

      5. hypot-def 3.1%

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

      6. fma-def 3.1%

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

      7. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 16.6%

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

   6. Taylor expanded in x around 0 79.4%

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

      1. distribute-rgt-in 79.4%

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

      2. *-un-lft-identity 79.4%

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

      3. associate-+l- 79.4%

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

      4. sub-neg 79.4%

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

      5. metadata-eval 79.4%

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

   8. Applied egg-rr 79.4%

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

4. Final simplification 94.0%

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

Alternative 6: 43.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + -1\\ t_1 := 1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{if}\;y \leq 1.05 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-223}:\\ \;\;\;\;t_0 \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-1 + \frac{x}{y} \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) -1.0)) (t_1 (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))))
   (if (<= y 1.05e-238)
     t_1
     (if (<= y 1.3e-223)
       (* t_0 (+ (/ x y) 1.0))
       (if (<= y 2.6e-139) t_1 (+ (/ x y) (+ -1.0 (* (/ x y) t_0))))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + -1.0;
	double t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)));
	double tmp;
	if (y <= 1.05e-238) {
		tmp = t_1;
	} else if (y <= 1.3e-223) {
		tmp = t_0 * ((x / y) + 1.0);
	} else if (y <= 2.6e-139) {
		tmp = t_1;
	} else {
		tmp = (x / y) + (-1.0 + ((x / y) * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y) + (-1.0d0)
    t_1 = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    if (y <= 1.05d-238) then
        tmp = t_1
    else if (y <= 1.3d-223) then
        tmp = t_0 * ((x / y) + 1.0d0)
    else if (y <= 2.6d-139) then
        tmp = t_1
    else
        tmp = (x / y) + ((-1.0d0) + ((x / y) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + -1.0;
	double t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)));
	double tmp;
	if (y <= 1.05e-238) {
		tmp = t_1;
	} else if (y <= 1.3e-223) {
		tmp = t_0 * ((x / y) + 1.0);
	} else if (y <= 2.6e-139) {
		tmp = t_1;
	} else {
		tmp = (x / y) + (-1.0 + ((x / y) * t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + -1.0
	t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)))
	tmp = 0
	if y <= 1.05e-238:
		tmp = t_1
	elif y <= 1.3e-223:
		tmp = t_0 * ((x / y) + 1.0)
	elif y <= 2.6e-139:
		tmp = t_1
	else:
		tmp = (x / y) + (-1.0 + ((x / y) * t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + -1.0)
	t_1 = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))))
	tmp = 0.0
	if (y <= 1.05e-238)
		tmp = t_1;
	elseif (y <= 1.3e-223)
		tmp = Float64(t_0 * Float64(Float64(x / y) + 1.0));
	elseif (y <= 2.6e-139)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + Float64(-1.0 + Float64(Float64(x / y) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + -1.0;
	t_1 = 1.0 + (-2.0 * ((y / x) * (y / x)));
	tmp = 0.0;
	if (y <= 1.05e-238)
		tmp = t_1;
	elseif (y <= 1.3e-223)
		tmp = t_0 * ((x / y) + 1.0);
	elseif (y <= 2.6e-139)
		tmp = t_1;
	else
		tmp = (x / y) + (-1.0 + ((x / y) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.05e-238], t$95$1, If[LessEqual[y, 1.3e-223], N[(t$95$0 * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e-139], t$95$1, N[(N[(x / y), $MachinePrecision] + N[(-1.0 + N[(N[(x / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.0500000000000001e-238 or 1.3e-223 < y < 2.5999999999999998e-139

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. associate-*r/ 66.4%

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

      3. +-commutative 66.4%

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

      4. fma-def 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in y around 0 29.3%

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

      1. unpow2 29.3%

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

      2. unpow2 29.3%

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

      3. times-frac 36.7%

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

   7. Applied egg-rr 36.7%

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

   if 1.0500000000000001e-238 < y < 1.3e-223

   1. Initial program 0.0%

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

      1. fma-def 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 3.1%

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

      4. fma-def 3.1%

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

      5. hypot-def 3.1%

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

      6. fma-def 3.1%

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

      7. hypot-def 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0 80.5%

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

   6. Taylor expanded in x around 0 80.0%

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

   if 2.5999999999999998e-139 < y

   1. Initial program 100.0%

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

      1. fma-def 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. times-frac 100.0%

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

      4. fma-def 100.0%

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

      5. hypot-def 100.0%

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

      6. fma-def 100.0%

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

      7. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 84.3%

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

   6. Taylor expanded in x around 0 84.0%

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

      1. distribute-rgt-in 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. associate-+l- 84.0%

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

      4. sub-neg 84.0%

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

      5. metadata-eval 84.0%

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

   8. Applied egg-rr 84.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-238}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-223}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-1 + \frac{x}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-238}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-223} \lor \neg \left(y \leq 3.1 \cdot 10^{-172}\right):\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-238)
   1.0
   (if (or (<= y 1.3e-223) (not (<= y 3.1e-172)))
     (* (+ (/ x y) -1.0) (+ (/ x y) 1.0))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-238) {
		tmp = 1.0;
	} else if ((y <= 1.3e-223) || !(y <= 3.1e-172)) {
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.05d-238) then
        tmp = 1.0d0
    else if ((y <= 1.3d-223) .or. (.not. (y <= 3.1d-172))) then
        tmp = ((x / y) + (-1.0d0)) * ((x / y) + 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-238) {
		tmp = 1.0;
	} else if ((y <= 1.3e-223) || !(y <= 3.1e-172)) {
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e-238:
		tmp = 1.0
	elif (y <= 1.3e-223) or not (y <= 3.1e-172):
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-238)
		tmp = 1.0;
	elseif ((y <= 1.3e-223) || !(y <= 3.1e-172))
		tmp = Float64(Float64(Float64(x / y) + -1.0) * Float64(Float64(x / y) + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e-238)
		tmp = 1.0;
	elseif ((y <= 1.3e-223) || ~((y <= 3.1e-172)))
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e-238], 1.0, If[Or[LessEqual[y, 1.3e-223], N[Not[LessEqual[y, 3.1e-172]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e-238 or 1.3e-223 < y < 3.1000000000000003e-172

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. associate-*r/ 65.4%

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

      3. +-commutative 65.4%

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

      4. fma-def 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in x around inf 34.2%

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

   if 1.0500000000000001e-238 < y < 1.3e-223 or 3.1000000000000003e-172 < y

   1. Initial program 88.7%

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

      1. fma-def 88.7%

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

      2. add-sqr-sqrt 88.7%

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

      3. times-frac 89.0%

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

      4. fma-def 89.0%

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

      5. hypot-def 89.0%

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

      6. fma-def 89.0%

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

      7. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 80.1%

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

   6. Taylor expanded in x around 0 79.8%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-238}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-223} \lor \neg \left(y \leq 3.1 \cdot 10^{-172}\right):\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-238} \lor \neg \left(y \leq 1.3 \cdot 10^{-223}\right) \land y \leq 8.2 \cdot 10^{-140}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.05e-238) (and (not (<= y 1.3e-223)) (<= y 8.2e-140)))
   (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))
   (* (+ (/ x y) -1.0) (+ (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1.05e-238) || (!(y <= 1.3e-223) && (y <= 8.2e-140))) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 1.05d-238) .or. (.not. (y <= 1.3d-223)) .and. (y <= 8.2d-140)) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else
        tmp = ((x / y) + (-1.0d0)) * ((x / y) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 1.05e-238) || (!(y <= 1.3e-223) && (y <= 8.2e-140))) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1.05e-238) or (not (y <= 1.3e-223) and (y <= 8.2e-140)):
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	else:
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 1.05e-238) || (!(y <= 1.3e-223) && (y <= 8.2e-140)))
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(Float64(Float64(x / y) + -1.0) * Float64(Float64(x / y) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 1.05e-238) || (~((y <= 1.3e-223)) && (y <= 8.2e-140)))
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	else
		tmp = ((x / y) + -1.0) * ((x / y) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.05e-238], And[N[Not[LessEqual[y, 1.3e-223]], $MachinePrecision], LessEqual[y, 8.2e-140]]], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e-238 or 1.3e-223 < y < 8.2000000000000003e-140

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. associate-*r/ 66.4%

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

      3. +-commutative 66.4%

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

      4. fma-def 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in y around 0 29.3%

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

      1. unpow2 29.3%

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

      2. unpow2 29.3%

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

      3. times-frac 36.7%

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

   7. Applied egg-rr 36.7%

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

   if 1.0500000000000001e-238 < y < 1.3e-223 or 8.2000000000000003e-140 < y

   1. Initial program 88.6%

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

      1. fma-def 88.6%

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

      2. add-sqr-sqrt 88.6%

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

      3. times-frac 89.0%

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

      4. fma-def 89.0%

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

      5. hypot-def 89.0%

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

      6. fma-def 89.0%

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

      7. hypot-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 83.9%

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

   6. Taylor expanded in x around 0 83.6%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-238} \lor \neg \left(y \leq 1.3 \cdot 10^{-223}\right) \land y \leq 8.2 \cdot 10^{-140}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-238}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-223}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-238)
   1.0
   (if (<= y 1.3e-223) -1.0 (if (<= y 4e-172) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-238) {
		tmp = 1.0;
	} else if (y <= 1.3e-223) {
		tmp = -1.0;
	} else if (y <= 4e-172) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.05d-238) then
        tmp = 1.0d0
    else if (y <= 1.3d-223) then
        tmp = -1.0d0
    else if (y <= 4d-172) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-238) {
		tmp = 1.0;
	} else if (y <= 1.3e-223) {
		tmp = -1.0;
	} else if (y <= 4e-172) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e-238:
		tmp = 1.0
	elif y <= 1.3e-223:
		tmp = -1.0
	elif y <= 4e-172:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-238)
		tmp = 1.0;
	elseif (y <= 1.3e-223)
		tmp = -1.0;
	elseif (y <= 4e-172)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e-238)
		tmp = 1.0;
	elseif (y <= 1.3e-223)
		tmp = -1.0;
	elseif (y <= 4e-172)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e-238], 1.0, If[LessEqual[y, 1.3e-223], -1.0, If[LessEqual[y, 4e-172], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e-238 or 1.3e-223 < y < 4.0000000000000002e-172

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. associate-*r/ 65.4%

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

      3. +-commutative 65.4%

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

      4. fma-def 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in x around inf 34.2%

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

   if 1.0500000000000001e-238 < y < 1.3e-223 or 4.0000000000000002e-172 < y

   1. Initial program 88.7%

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

      1. +-commutative 88.7%

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

      2. associate-*r/ 88.6%

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

      3. +-commutative 88.6%

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

      4. fma-def 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in x around 0 78.9%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-238}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-223}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-*r/ 71.0%

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

   3. +-commutative 71.0%

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

   4. fma-def 71.0%

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

3. Simplified 71.0%

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

5. Taylor expanded in x around 0 69.4%

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

6. Final simplification 69.4%

   \[\leadsto -1 \]
7. Add Preprocessing

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024018 
(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))))