Kahan p9 Example

Percentage Accurate: 68.6% → 100.0%

Time: 8.0s

Alternatives: 10

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

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 67.5%

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

   1. add-sqr-sqrt 67.5%

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

   2. times-frac 67.5%

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

   3. hypot-def 67.5%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. 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 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

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 67.5%

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

   1. add-sqr-sqrt 67.5%

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

   2. times-frac 67.5%

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

   3. hypot-def 67.5%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 92.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (fma 2.0 (* (/ 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 = fma(2.0, ((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 = fma(2.0, Float64(Float64(x / y) * Float64(x / y)), -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[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   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-*r/ 3.1%

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

      2. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 55.4%

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

      1. fma-neg 55.4%

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

      2. unpow2 55.4%

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

      3. unpow2 55.4%

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

      4. times-frac 77.5%

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

      5. metadata-eval 77.5%

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

   6. Simplified 77.5%

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

4. Final simplification 92.7%

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

Alternative 4: 92.5% accurate, 0.5× speedup

Math

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

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

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) / y) / (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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = ((x + y) / y) / (y / (x - y))
    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) / y) / (y / (x - y));
	}
	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) / y) / (y / (x - y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x + y) / y) / Float64(y / Float64(x - y)));
	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) / y) / (y / (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] / N[(y / N[(x - y), $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 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-*r/ 3.1%

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

      2. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. unpow2 3.1%

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

   6. Simplified 3.1%

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

      1. clear-num 3.1%

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

      2. un-div-inv 3.1%

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

   8. Applied egg-rr 3.1%

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

      1. associate-/l* 0.0%

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

      2. times-frac 77.1%

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

   10. Simplified 77.1%

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

       1. *-commutative 77.1%

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

       2. clear-num 77.1%

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

       3. un-div-inv 77.1%

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

   12. Applied egg-rr 77.1%

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

4. Final simplification 92.5%

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

Alternative 5: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162} \lor \neg \left(y \leq 5.6 \cdot 10^{-164}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.1e-24)
   -1.0
   (if (or (<= y -1.6e-162) (not (<= y 5.6e-164)))
     (* (- x y) (/ (+ x y) (+ (* x x) (* y y))))
     (+ 1.0 (* -2.0 (* (/ y x) (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.1e-24) {
		tmp = -1.0;
	} else if ((y <= -1.6e-162) || !(y <= 5.6e-164)) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.1d-24)) then
        tmp = -1.0d0
    else if ((y <= (-1.6d-162)) .or. (.not. (y <= 5.6d-164))) then
        tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
    else
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.1e-24) {
		tmp = -1.0;
	} else if ((y <= -1.6e-162) || !(y <= 5.6e-164)) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.1e-24:
		tmp = -1.0
	elif (y <= -1.6e-162) or not (y <= 5.6e-164):
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
	else:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.1e-24)
		tmp = -1.0;
	elseif ((y <= -1.6e-162) || !(y <= 5.6e-164))
		tmp = Float64(Float64(x - y) * Float64(Float64(x + y) / Float64(Float64(x * x) + Float64(y * y))));
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.1e-24)
		tmp = -1.0;
	elseif ((y <= -1.6e-162) || ~((y <= 5.6e-164)))
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	else
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.1e-24], -1.0, If[Or[LessEqual[y, -1.6e-162], N[Not[LessEqual[y, 5.6e-164]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.10000000000000036e-24

   1. Initial program 53.1%

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

      1. associate-*r/ 54.3%

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

      2. fma-def 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if -6.10000000000000036e-24 < y < -1.59999999999999988e-162 or 5.6000000000000002e-164 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 96.8%

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

      2. 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 -1.59999999999999988e-162 < y < 5.6000000000000002e-164

   1. Initial program 53.8%

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

      1. associate-*r/ 54.5%

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

      2. fma-def 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in y around 0 53.8%

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

      1. unpow2 53.8%

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

      2. unpow2 53.8%

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

   6. Simplified 53.8%

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

      1. times-frac 79.1%

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

   8. Applied egg-rr 79.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-162} \lor \neg \left(y \leq 5.6 \cdot 10^{-164}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 6: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-169} \lor \neg \left(y \leq 9.5 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55e-169) (not (<= y 9.5e-153)))
   (+ (* (/ x y) (/ x y)) -1.0)
   (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e-169) || !(y <= 9.5e-153)) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.55d-169)) .or. (.not. (y <= 9.5d-153))) then
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e-169) || !(y <= 9.5e-153)) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.55e-169) or not (y <= 9.5e-153):
		tmp = ((x / y) * (x / y)) + -1.0
	else:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55e-169) || !(y <= 9.5e-153))
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.55e-169) || ~((y <= 9.5e-153)))
		tmp = ((x / y) * (x / y)) + -1.0;
	else
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55e-169], N[Not[LessEqual[y, 9.5e-153]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5500000000000001e-169 or 9.50000000000000031e-153 < y

   1. Initial program 72.2%

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

      1. associate-*r/ 72.3%

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

      2. fma-def 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in x around 0 58.3%

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

      1. unpow2 58.3%

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

   6. Simplified 58.3%

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

      1. clear-num 58.4%

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

      2. un-div-inv 58.5%

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

   8. Applied egg-rr 58.5%

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

      1. associate-/l* 58.3%

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

      2. times-frac 86.0%

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

   10. Simplified 86.0%

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

   11. Taylor expanded in x around 0 84.8%

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

       1. sub-neg 84.8%

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

       2. unpow2 84.8%

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

       3. unpow2 84.8%

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

       4. metadata-eval 84.8%

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

   13. Simplified 84.8%

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

       1. times-frac 86.0%

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

   15. Applied egg-rr 86.0%

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

   if -1.5500000000000001e-169 < y < 9.50000000000000031e-153

   1. Initial program 57.8%

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

      1. associate-*r/ 57.0%

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

      2. fma-def 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in y around 0 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

   6. Simplified 55.2%

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

      1. times-frac 79.5%

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

   8. Applied egg-rr 79.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-169} \lor \neg \left(y \leq 9.5 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 7: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-169}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-153}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{y}}{\frac{y}{x - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.95e-169)
   (+ (* (/ x y) (/ x y)) -1.0)
   (if (<= y 7.8e-153)
     (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))
     (/ (/ (+ x y) y) (/ y (- x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.95e-169) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else if (y <= 7.8e-153) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = ((x + y) / y) / (y / (x - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.95d-169)) then
        tmp = ((x / y) * (x / y)) + (-1.0d0)
    else if (y <= 7.8d-153) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else
        tmp = ((x + y) / y) / (y / (x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.95e-169) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else if (y <= 7.8e-153) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = ((x + y) / y) / (y / (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.95e-169:
		tmp = ((x / y) * (x / y)) + -1.0
	elif y <= 7.8e-153:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	else:
		tmp = ((x + y) / y) / (y / (x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.95e-169)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + -1.0);
	elseif (y <= 7.8e-153)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(Float64(Float64(x + y) / y) / Float64(y / Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.95e-169)
		tmp = ((x / y) * (x / y)) + -1.0;
	elseif (y <= 7.8e-153)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	else
		tmp = ((x + y) / y) / (y / (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.95e-169], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, 7.8e-153], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] / N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.94999999999999988e-169

   1. Initial program 64.4%

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

      1. associate-*r/ 64.6%

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

      2. fma-def 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in x around 0 54.5%

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

      1. unpow2 54.5%

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

   6. Simplified 54.5%

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

      1. clear-num 54.6%

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

      2. un-div-inv 54.6%

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

   8. Applied egg-rr 54.6%

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

      1. associate-/l* 54.3%

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

      2. times-frac 89.8%

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

   10. Simplified 89.8%

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

   11. Taylor expanded in x around 0 88.4%

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

       1. sub-neg 88.4%

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

       2. unpow2 88.4%

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

       3. unpow2 88.4%

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

       4. metadata-eval 88.4%

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

   13. Simplified 88.4%

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

       1. times-frac 89.9%

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

   15. Applied egg-rr 89.9%

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

   if -1.94999999999999988e-169 < y < 7.8000000000000004e-153

   1. Initial program 57.8%

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

      1. associate-*r/ 57.0%

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

      2. fma-def 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in y around 0 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

   6. Simplified 55.2%

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

      1. times-frac 79.5%

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

   8. Applied egg-rr 79.5%

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

   if 7.8000000000000004e-153 < y

   1. Initial program 99.8%

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

      1. associate-*r/ 99.5%

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

      2. 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 72.0%

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

      1. unpow2 72.0%

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

   6. Simplified 72.0%

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

      1. clear-num 72.0%

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

      2. un-div-inv 72.2%

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

   8. Applied egg-rr 72.2%

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

      1. associate-/l* 72.3%

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

      2. times-frac 72.3%

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

   10. Simplified 72.3%

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

       1. *-commutative 72.3%

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

       2. clear-num 72.3%

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

       3. un-div-inv 72.3%

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

   12. Applied egg-rr 72.3%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-169}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-153}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{y}}{\frac{y}{x - y}}\\ \end{array} \]

Alternative 8: 83.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-170} \lor \neg \left(y \leq 1.52 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.9e-170) (not (<= y 1.52e-152)))
   (+ (* (/ x y) (/ x y)) -1.0)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.9e-170) || !(y <= 1.52e-152)) {
		tmp = ((x / y) * (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 <= (-4.9d-170)) .or. (.not. (y <= 1.52d-152))) then
        tmp = ((x / y) * (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 <= -4.9e-170) || !(y <= 1.52e-152)) {
		tmp = ((x / y) * (x / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.9e-170) or not (y <= 1.52e-152):
		tmp = ((x / y) * (x / y)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.9e-170) || !(y <= 1.52e-152))
		tmp = Float64(Float64(Float64(x / y) * 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 <= -4.9e-170) || ~((y <= 1.52e-152)))
		tmp = ((x / y) * (x / y)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.9e-170], N[Not[LessEqual[y, 1.52e-152]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -4.8999999999999996e-170 or 1.52000000000000009e-152 < y

   1. Initial program 72.2%

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

      1. associate-*r/ 72.3%

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

      2. fma-def 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in x around 0 58.3%

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

      1. unpow2 58.3%

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

   6. Simplified 58.3%

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

      1. clear-num 58.4%

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

      2. un-div-inv 58.5%

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

   8. Applied egg-rr 58.5%

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

      1. associate-/l* 58.3%

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

      2. times-frac 86.0%

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

   10. Simplified 86.0%

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

   11. Taylor expanded in x around 0 84.8%

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

       1. sub-neg 84.8%

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

       2. unpow2 84.8%

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

       3. unpow2 84.8%

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

       4. metadata-eval 84.8%

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

   13. Simplified 84.8%

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

       1. times-frac 86.0%

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

   15. Applied egg-rr 86.0%

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

   if -4.8999999999999996e-170 < y < 1.52000000000000009e-152

   1. Initial program 57.8%

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

      1. associate-*r/ 57.0%

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

      2. fma-def 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in x around inf 78.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-170} \lor \neg \left(y \leq 1.52 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 82.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-151}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e-169) -1.0 (if (<= y 5.5e-151) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.6e-169) {
		tmp = -1.0;
	} else if (y <= 5.5e-151) {
		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 <= (-1.6d-169)) then
        tmp = -1.0d0
    else if (y <= 5.5d-151) 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 <= -1.6e-169) {
		tmp = -1.0;
	} else if (y <= 5.5e-151) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.6e-169:
		tmp = -1.0
	elif y <= 5.5e-151:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.6e-169)
		tmp = -1.0;
	elseif (y <= 5.5e-151)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.6e-169)
		tmp = -1.0;
	elseif (y <= 5.5e-151)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.6e-169], -1.0, If[LessEqual[y, 5.5e-151], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.59999999999999997e-169 or 5.4999999999999998e-151 < y

   1. Initial program 72.2%

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

      1. associate-*r/ 72.3%

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

      2. fma-def 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in x around 0 85.1%

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

   if -1.59999999999999997e-169 < y < 5.4999999999999998e-151

   1. Initial program 57.8%

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

      1. associate-*r/ 57.0%

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

      2. fma-def 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in x around inf 78.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-151}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 66.5% 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 67.5%

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

   1. associate-*r/ 67.3%

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

   2. fma-def 67.3%

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

3. Simplified 67.3%

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

4. Taylor expanded in x around 0 64.4%

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

5. Final simplification 64.4%

   \[\leadsto -1 \]

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 2023178 
(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))))